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Full text of "A modern arithmetic, with graphic & practical exercises"

ARITHMETIC & PRACTICAL EXERCISES H.SYDNEY JONI--S WITH ANSWERS IN MEMORIAM FLORIAN CAJORI "I A y A MODERN ARITHMETIC MACMILLAN AND CO-, LIMITED LONDON BOMBAY CALCUTTA MELBOURNE THE MACMILLAN COMPANY NEW YORK BOSTON CHICAGO ATLANTA SAN FRANCISCO THE MACMILLAN CO. OF CANADA, LTD TORONTO A MODERN ARITHMETIC WITH GRAPHIC & PRACTICAL EXERCISES BY H. SYDNEY JONES, M.A. i / LATE SCHOLAR OF CHRIST'S COLLEGE, CAMBRIDGE HEADMASTER OF CHELTENHAM GRAMMAR SCHOOL MACMILLAN AND CO., LIMITED ST. MARTIN'S STREET, LONDON 1909 First Edition 1908. Reprinted 1909. GLASGOW : HRINTKD AT THE UNIVEKSITY PRESS HV ROBERT MACLEHOSE AND CO. LTD. PREFACE DURING the past few years various suggestions for the improved teaching of arithmetic have been made by the Board of Education, by committees of the British Association and the Mathematical Association, and by some public examin- ing bodies. Though arithmetic was included in the school curriculum chiefly for utilitarian reasons, the abuse of exam- inations led to the introduction of tiresome and complicated calculations of no service in practical life and of little educa- tional value. These undesirable developments are, however, now disappearing, and, in common with other mathematical subjects, the scope, substance and method of arithmetic are being placed upon a more reasonable basis. In this book an attempt has been made to provide a course of work in which the essential parts of arithmetic are dealt with in the spirit of recent recommendations. The author has endeavoured to produce a practicable as well as a practical course which mathematical teachers may introduce without any violent dislocation of existing practice and with the knowledge that the requirements of the chief examining bodies, which determine to a large extent the nature of the teaching in schools, have not been disregarded. Any efficient teaching of arithmetic must aim at (i) a clear conception of units of quantities involved in calculations, (2) accuracy, (3) quickness vi PREFACE in the manipulation of numbers, (4) cultivation of the reason- ing faculties. These principles have determined the character of the exercises throughout the book. Among other distinguishing features of the volume may be mentioned : (1) Decimals are introduced at an early stage and before vulgar fractions are explained, with a view to teach students to think in decimals. This introduction is effected by means of practical exercises in measurement in the Metric System and by a judicious extension of the idea of Place Value. (2) Practical measurements and other exercises, which can be done in ordinary class rooms, are provided at con- venient intervals throughout the book and are accompanied by numerous graphic exercises, where these are likely to assist the formation of correct ideas. All exercises have been designed to provide material for mathematical deduction, and will, it is believed, convey sound notions of concrete quantities. These sections of the book are so arranged, however, that the practical exercises can be omitted without any loss of continuity. (3) The arithmetical exercises illustrate in most cases actual calculations met with in the experience of life. Easy illustrative and stimulating problems precede those which are purely numerical ; many of the latter are included to train the student to become able to perform the fundamental operations of arithmetic in a sub-conscious manner. (4) Mental and Oral exercises, intended to introduce new ideas and to stimulate the habit of ready calculation, are included in each Section. (5) Early exercises work out exactly ; but, as the knowledge of the student extends, approximations are introduced, and he is led in various ways to know how to estimate the degree of accuracy possible from the data with which he is provided. PREFACE vii In this way approximate methods of working lead, not to loose, but to more accurate trains of thought. (6) Graded Revision Exercises are inserted at various stages of progress, and specimen Examination Papers are included to show the student what may be expected of him by public examining bodies. (7) The practical, graphic and purely arithmetical exercises are arranged on one common and logical plan, with the view of giving a concrete and connected knowledge of numerical calculations. No apology is necessary for the introduction of a section on averages and elementary statistics ; and it is hoped that the treatment of Stocks may make that subject more real and interesting. Probably the graphic method of calculating volumes and the use of logarithmic graphs have not been used in all schools ; some teachers will be surprised at the accuracy of the former and the field of application of both methods. Part I. covers the subjects prescribed for the Oxford and Cambridge Junior Local examinations in arithmetic, and other examinations of about the same scope : the Second Part will be found to meet, not only the requirements of the more conservative examining bodies, but also those of the Army, Civil Service and other examiners keenly alive to the change taking place in mathematical teaching. The author wishes here to express his thanks to the various examining bodies which have courteously sanctioned the use of their examination papers, and in particular the Oxford Local Delegates, the Cambridge Local Syndics, the Senates of London and Birmingham Universities, the College of Pre- ceptors, the London Chamber of Commerce, and the Con- troller of H.M. Stationery Office, in whom is vested the viii PREFACE copyright of papers set by the Civil Service Commissioners and the Board of Education. It is very difficult for the author adequately to express his gratitude to Prof. R. A. Gregory and Mr, A. T. Simmons for the most unremitting care and thought which they have expended on the MSS. and proof sheets of this work ; while he must add that the conception of the work was theirs in the first instance. H. SYDNEY JONES. CHELTENHAM, August 1908. CONTENTS PART I SECTION PAGE 1. NOTATION AND NUMERATION, ..... i 2. ADDITION, 4 Use of Signs. Addition by Steps. Addition of Long Columns. 3. SUBTRACTION, 12 4. MULTIPLICATION, 18 Multiplication by single digits. Multiplication by factors. General Case. Squared and Cubed Numbers. 5. DIVISION, 29 Division by Factors. Division by the General Method. Italian Method. Division true to the Nearest Unit. 6. FACTORS, .... 37 7. ARITHMETICAL AVERAGES, 40 8. BRITISH MONEY SYSTEMS, 53 9. COMPOUND ADDITION, ... ... 56 10. COMPOUND SUBTRACTION, ...... 58 11. COMPOUND MULTIPLICATION, 61 12. COMPOUND DIVISION, 65 x CONTENTS SECTION PAGE 13. THE MEASUREMENT OF LENGTH, .... 74 14. REDUCTION, 75 15. ADDITION, SUBTRACTION, MULTIPLICATION AND DIVISION OF LENGTHS, 79 1 6. REDUCTION IN THE METRIC OR DECIMAL SYSTEM, . 85 17. INTRODUCTION OF DECIMAL NOTATION, ... 88 18. ADDITION AND SUBTRACTION OF METRIC LENGTHS, 91 19. MULTIPLICATION AND DIVISION OF DECIMALS BY A SINGLE FIGURE, 95 20. MULTIPLICATION OF DECIMALS, 98 General Case. 21. DIVISION OF DECIMALS, . ... 107 General Case. 22. DIVISION TRUE TO A CERTAIN NUMBER OF DECIMAL PLACES, no 23. BRITISH MEASURES OF LENGTH (Continued), . .113 24. AREA, 116 25. BRITISH MEASURES OF AREA, . . . . . 117 Units of Area. 26. METRIC MEASURES OF AREA, 118 Measurement of Irregular Areas. 27. DECIMAL COINAGE, . . 130 28. VOLUMES, 132 29. PROBLEMS ON MEASURES OF CAPACITY, . . .139 30. MEASUREMENT OF WEIGHT, 141 31. MEASUREMENT OF TIME, 147 CONTENTS xi SECTION PAGE 32. VULGAR FRACTIONS, 157 Graphic Illustrations of Vulgar Fractions. 33. IMPROPER FRACTIONS AND MIXED NUMBERS, . . 160 One Quantity as a Vulgar Fraction of another. 34. EQUIVALENT FRACTIONS, 162 Equivalent Fractions Illustrated Graphically. 35. ADDITION AND SUBTRACTION OF FRACTIONS, . .166 36. CONVERSION OF FRACTIONS, 169 Conversion of a Decimal into a Vulgar Fraction. Conver- sion of a Vulgar Fraction into a Decimal. Recurring Decimals. 37. MULTIPLICATION AND DIVISION OF VULGAR FRAC- TIONS, 172 Multiplication by a Whole Number. Division by a Whole Number. Multiplication of one Fraction by another. Division of one Fraction by another. 38. GREATEST COMMON MEASURE, 182 Graphical Introduction to General Method. General Method. 39. LEAST COMMON MULTIPLE, 191 General Method. 40. PROPORTION, 197 41. PERCENTAGES, 207 Scale Drawing. 42. CALCULATIONS AND APPROXIMATIONS, . . . 236 Decimalization of Money. Possible Errors in Decimaliza- tion. 43. PRACTICE, 244 xii CONTENTS SECTION PAGE 44. ERRORS IN DATA, 254 45. APPROXIMATE OR CONTRACTED MULTIPLICATION, . 259 46. CONTRACTED DIVISION, 268 47. SQUARE ROOT, 273 General Method. Square Roots of Vulgar Fractions. 48. INTEREST, ......... 278 'Simple Interest. REVISION PAPERS, ... . 289 PART II PAGE 49. MEASUREMENT OF LENGTH (Continued) . . .297 50. THE USE OF THE CHAIN . . . ,,..;. -303 51. APPROXIMATIONS (Continued} 307 Approximations in Multiplication. Approximations in Division. Approximations in the Extraction of Square Root. Surds. Approximations in Square Root (Con- tinued]. Square Root (Continued). 52. PROPORTION (Continued) ...... 323 53. MEASUREMENT OF ANGLES 327 54. AREAS OF CIRCLES 331 55. THE PRISM AND CYLINDER 334 56. SIMILAR FIGURES . . . . . . .338 57. INVERSE PROPORTION 342 Examination of Quantities for Proportion. 58. AREAS OF RECTILINEAR FIGURES . . 358 59. LOGARITHMS 387 60. LOGARITHMIC TABLES AND INSTRUMENTS . . . 396 Evaluation of Formulae. The Slide Rule. xiv CONTENTS SECTION PAGE 61. COMPOUND INTEREST 414 The Compound Interest Law. Geometrical Progression. 62. THE AREA OF A TRIANGLE 432 - 63. BILLS OF EXCHANGE AND BANKER'S DISCOUNT . . 436 64. CALCULATION OF AREAS (Continued) .... 438 Spherical Surfaces. 65. CALCULATION OF VOLUMES (Continued) . . .447 Volumes of Cones and Pyramids. Volumes of Frusta. Volumes of Spheres. Graphic Calculation of Volumes. "** 66. EQUATION OF PAYMENTS 478 _ 67. FOREIGN EXCHANGES 482 Money Exchange between different Countries. 68. FOREIGN BILLS OF EXCHANGE 487 69. STOCKS AND SHARES 490 Brokerage and other Charges. 70. STATISTICS . .505 Averages. Errors in Practical Work. Method of Mean Squares. Permutations and Combinations. Probabilities. Expectation of Life. TYPICAL EXAMINATION PAPERS 555 TABLES 585 ANSWERS 599 INDEX 659 A MODERN ARITHMETIC 1. NOTATION AND NUMERATION. Arithmetic is the science treating of numbers and calcu- lations. Numbers are expressed by means of certain signs or symbols. These are, in the very great majority of cases, figures ; and occasionally the capital letters I, V, X, L, C, D, M are used. Calculations are always made with figures. Figures, or digits, i, 2, 3, 4, 5, 6, 7, 8, 9, represent one, two, three, four, five, six, seven, eight, nine units respectively. The fact of our figures being often called digits (Latin, digitus, a finger) suggests that no doubt the primitive way of expressing numbers up to ten was by means of the fingers of the two hands. The letters which are occasionally used to represent numbers are : M, a thousand ; D, five hundred ; C, one hundred ; L, fifty ; X, ten ; V, five ; and I, one. The basis of all calculation is the unit i, one ; and since there are only the above symbols to represent numbers, it is evident that in order to express numbers exceeding 9 it is necessary to give a place value or local value to the digits ; that is to make the value of a digit depend upon position as well as upon the symbol. J.M.A. A 2 A MODERN ARITHMETIC Figures have two values an intrinsic one, and a place value when arranged to form numbers. So, too, has the symbol I in the case of letters. The symbol o naught, nought, or cypher has neither an intrinsic nor a place value, but is introduced in expressing the place value of the other digits. Letters used to represent numbers have no place value, excepting I and X. I or X, placed to the left of a symbol, signifies that the number represented is short by that amount, while, if placed to the right, signifies that the number repre- sented is greater by the amount. Thus IV, VI, VIII, IX, and XII represent severally 4, 6, 8, 9, 12, etc. XC is 90. A figure in the first place of a number denotes so many units, and the same figure in the second place (proceeding from right to left) denotes 10 times as many. Hence, figures in the second place are often called Tens. A figure in the third place represents ten times as many as it would in the second place, or a hundred times as many as it would in the first place. These third place figures are called Hundreds. Similarly, figures in the fourth place are called Thousands; in the fifth place, Tens of Thousands; in the sixth place, Hundreds of Thousands. When we come to the seventh place a figure there represents Millions. The million is often taken as a new kind of unit, and a repetition of the former process then occurs. A million millions is called in Great Britain a billion, beyond which few numbers are ever taken. (In France and the United States a billion is taken to be one thousand millions.) The notation and numeration table up to a million would then be as on the next page. It will be noticed that a figure in occupying places successively one place to the left represents a value ten times greater than before. NOTATION AND NUMERATION COMMON SYSTEM OF NOTATION. 1 2 en o 'c H $ Numbers expressed in 8 jl Words. rf 1 p c "8 e 1 5 c rf jj r 3 2 a E "c s ffi H H ffi H D 9 Nine. i Ten. 9 9 Ninety nine. i o One hundred. 9 9 9 Nine hundred and ninety nine. i One thousand. / Nine thousand, nine hundred and 9 9 9 9 \^ ninety nine. i o o o o Ten thousand. 9 9 9 9 9 /Ninety nine thousand, nine hundred ( and ninety nine. I o o o o One hundred thousand. 9 9 9 9 9 9 (Nine hundred and ninety nine thousand, ^ nine hundred and ninety nine. 1 o One million. It is customary and advantageous to divide or separate the figures expressing a given number by a comma into periods of three, thus 3,465,908 Notation is the method of expressing a number given in words by means of figures, and Numeration is the method of writing a number in words when represented by figures. When any value, as tens, thousands, etc., is not given, the place is supplied by cypher o. Thus, seven thousand and six would be written 7,006 4 A MODERN ARITHMETIC EXERCISES I. 1. Make a table like that on page 3, and show in it the numbers (a) 3 million 2 hundred and 6, (b) 71 thousand and twenty five, (c) 5 hundred thousand 2 hundred, {d} 9 thousand and 9. Express in numbers : 2. (a) Twenty, (b) twenty three, (c) fifteen, (d) ten, (e) eighteen, (_/") sixty two. 3. (a) Eighty six, (b) one hundred, (c) one hundred and one, (d) one hundred and ten, (e) one hundred and eleven. 4. (a) One hundred and twenty eight, (b) one hundred and eighty two, (c) one hundred and eight, (d) one hundred and eighty. 5. (a) Three hundred and three, (b) three hundred and thirty, (c) three hundred and thirty three. 6. (a) Two thousand and six, (b) six thousand and two, (c) six thousand and twenty, (d} six thousand two hundred. 7. (#) Eight thousand, seven hundred and six, (b) eight thousand and seventy six, (c) eight hundred and seventy six. Express in words : 8. (a) 3, (b) 8, (c) 1 8, (<*) 81, (e) 25, (/) 52, (^) 520, (h) 300, (0 3000, (j) 3005, () 3050, (/) 3500, (m) 5003, () 1 86, (o) 168, (^) 861, (?) 816, (r) 71, CO 701, (/) 7001. 2. ADDITION. Addition is the process of finding one number, or quantity, which is equivalent to two or more numbers or quantities. The simple numbers which are to be collected into one are often called terms, or addends, while the result obtained is called the Sum, or. Total. In finding the Sum of any number of terms it is usual to arrange them vertically, taking care that units figures are Th. ADDITION 5 vertically under one another, and so with the tens, and other digits. EXAMPLE. To find the sum 0/476, 3089, and 940. The numbers are arranged. The units added up make fifteen (15) which is i ten and 5 units. The five units are placed in the units column, and the one ten carried to the tens column. 1 The tens increased by this i make twenty tens (200), which is 2 hundreds and o tens. The cypher o is put in the tens column, and the 2 hundreds carried on to the hundreds, which become fifteen hundreds, or one (i) thousand and 5 hundreds. Placing the 5 under the hundreds and carrying the i thousand to the 3 thousands the total becomes 4 thousands. The sum or total of the 3 addends is 4505. It is by no means necessary to place the terms vertically under one another. They may be arranged horizontally, and then added up; Here care has to be taken that the units are added to units, the tens to the tens, and so on. In the process of addition the two essential requisites are speed and accuracy. To ensure speed it is a good practice to go through such exercises as the following. Start with a given number, say 7, and then continue to add say 3's, up to any given number, thus 7, 10, 13, 16, etc. The same may be done by adding on other numbers ; continuing from a given start, thus, 6, and adding on 7, we obtain 6, 13, 20, 27, and so on for any number of times. To secure accuracy, it is well to add the columns from bottom to top, and then from top to bottom ; or, when placed horizontally, to add the figures from left to right and then from right to left. If the totals agree in each case, it is very likely that the answer is correct. 6 A MODERN ARITHMETIC EXERCISES II. Find the sums of the following, checking your results by adding from bottom to top, and then from top to bottom. 1. 75 2. 963 3. 541 4. 829 382 875 326 314 971 314 715 692 64 65 493 508 28 802 654 344 394 68 19 182 5. A woman has three baskets full of eggs ; one basket contains 22 eggs, the second basket 30 eggs, and the third basket contains 19 eggs ; how many eggs are there altogether? 6. In a garden there are six apple trees, one with 107 apples on it, the second with 36 apples on it, the third with 76 apples, the fourth with 52 apples, the fifth with no apples and the sixth with 82 apples. How many apples does the garden contain ? 7. There are three bags of marbles, one containing 26 marbles, the second 57 marbles, and the third 48 marbles ; how many marbles are there altogether? Add: 8. 26, 212, 18, 77, 84. 9. 38, 146, 372, 237, 96 10. 345, 246, 348. 11. 318, 76, 208, 79. 12. 592, 685, 23, 82, 71. 13. 734, 72, 83, 174, 536. 14. 492, 218, 339, 297, 355, 210. 15. 34, 25, 36, 48, 348, 19, 7. 16. 207, 316, 450, 38, 27, 34. Find the totals of the following, checking your results by adding from right to left and from left to right. 17. 826, 45, 167, 45, 792, 75- 18. 957, 63, 14, 770, 845, 64. 19. 145, 896, 56, 549, 14, 342. 20. 567, 265, 26, 367, 129, 36, 267, 30. 21. 589, 742, 180, 246, 352, 705. ADDITION 7 EXERCISES III. Express in words the numbers represented by the figures marked with an asterisk (*). 1. 382. 2. 1082. 3. 2161. 4. 818. 5. 756- 6. 2 1 6*. 7. ^623. 8. 1885. 9. 326. 10. 236. 11. 1832. 12. 1832. 13. 1761. 14. 58 1. 15. 1013. * * 16. 101. 17. 1 10. Write down the sum of the numbers represented by the digits marked with an asterisk. * * * * 18. 8325, 2613, 1822, 2764. 19. 8325, 2613, 1822, 2764. 20. 8002, 82, 2008, 208. 21. 736, 817, 2926, 3184. 22. 763, 568, 921, 374. 23. 536, 230, 203, 1218. 24. 536, 230, 203, 1218. 25. 172, 172, 172. 26. 172, 712, 271. 27. 172, 712, 712. Use of Signs. The symbol + (plus) is used to express addition. Thus, 4 + 5 means 4 added to 5, and the answer is 9, and the state- ment that four added to five makes nine could be represented by making use of another symbol the equality symbol = . Thus, 4 + 5 = 9- The symbols .'. and v will also frequently be employed; .'. signifies therefore, because. EXEECISES IV. Evaluate, that is, find the value of: 1. 28 + 316 + 459 + 326+182. 2. 56 + 89 + 376 + 83 + 672. 3. 36+112 + 172 + 56 + 29. 4. 42 + 226 + 672 + 53 + 726. 8 A MODERN ARITHMETIC Evaluate, that is, find the value of : 5. 38+176 + 54+365 + 27. 6. 126+58+98+462 + 567. 7. 316 + 98 + 327+42 + 389. 8. 389 + 627 + 544+89+72. 9. 16 + 54+237 + 84+112. 10. 56+3276+81+8964. 11. 12 + 87 + 543 + 898 + 276+543. 12. 6+13 + 897 + 26 + 8679+345+6. 13. 856+983 + 72 + 543 + 876. 14. 32+97 + 856 + 321+765 + 50. 15. 3 + 895 + 726+86 + 5890+32. 16. 5432 + 674+52 + 1000 + 371. 17. 296 + 154+172 + 345 + 89+1654. Addition by Steps. The following are the results of borings for wells, the tables giving the thicknesses of the various sorts of strata met with ; add up and give the depths of the different strata below the surface, also the total depth of the well, as in the example (Well at Romford). EXAMPLE. Well at Romford : Oi , Thickne Soil - 7 7 Gravel 9 16 Blue clay 29 45 Stone - 2 47 Green sand - 18 65 Loamy sand 17 82 Bright green sand 20 102 Green sand - 32 134 Hard blue clay 18 152 ADDITION EXERCISES V. 1. WellatHamp Strata. Brown clay - Blue - Sandy - Basement bed Plastic clay - Sand - Hard chalk - Soft - - Hard - - 3. West Drayton Strata. T jj Mould Clay - Gravel Clay - Basement bed Blue clay Clay - * * Red clay Sand - Clay - stead : 2. Well at Walham G Strata. "^n'fert Soil - 6 Sand and Gravel - 1 5 Blue London clay 119 Mottled clay - 36 Sand - - - 4 Mottled clay - 15 Sand, stones, water 7 Brown sand and clay 35 Green 3 Flints - - - ' i Chalk, Flints, Rock chalk - - 1 70 4. Westminster, a through an old well : Strata. C fT Old well 118 Sand - 5 Clay - 10 Hard rock - 2 Sand and stone - 2 Sand and clay - 10 Shale 2 Mottled stone - 16 Green sand i Sand and pebbles 21 Grey sand 24 Flints - - i Chalk - - ' - 205 reen : 55 Depth. I7O 84 c 4O 40 4.O A 28 T. boring s Depth. 2 1 1 61 2 IO II g 7 28 6 io A MODERN ARITHMETIC 5. Wandsworth : *-*. T !ffeeT ^ Yellow clay - - 12 Blue - - 158 Black - - 20 Mottled - 41 Yellow - - 5 Green sand - - 36 Flints 2 Chalk - 60 Addition of Long Columns. In adding long sums it is frequently advisable to mark the hundreds as in the following example : EXAMPLE. Add 28, 76, 126, 92, 75, 38, 127, 36, 84, 59, 96, 74, 89, 438, 127, 399, 418, 385, 59. From Units Column. Tens. Hundreds 28 126 112 76 118 no 126 112 103 28 92 106 101 75, 104 92 38 99 85 Add up the units in the 127 91 82 27 usual way, the totals, as the ad- 36 84 80 dition proceeds, being marked 84 78 77 on the right; these however 59 74 69 should not be written down 96 65 64 until the hundred is reached, 74 59 55 when the 5 is ticked and 104 89 55 48 marked on the right. 438 46 40 26 The tens column is treated 127 38 37 22 similarly. 399 3i 35 21 418 22 26 18 385 14 25 14 59 17 2826 II, 12 ADDITION ii EXERCISES VI. Find the sum of the following groups, marking the hundreds, etc., as in the preceding example. 1. 36, 48, 32, 70, 83, 56, 34, 92, 57, 55, 32, 29, 38, 59, 78, 98, 8, 57, 96, 88, 57, 68, 39. 2. 99, 88, 78, 79, 67, 75, 58, 78, 49, 89, 96, 87, 98, 92, 75, 56, 57, 58, 199, 77. 3. 239, 259, 39, 57, 68, 79, 38, 36, 55, 66, 74, 85, 97, 97, 38, 56, 54, 187, 88, 94, 29, 37. 4. 57, 66, 78, 87, 84, 96, 58, 58, 67, 66, 28, 27, 39, 57, 86, 78, 93, 99, 98, 64, 67, 19, 18, 17, 55. 5. 86, 88, 95, 98, 97, 67, 76, 54, 47, 48, 49, 74, 89, 97, 96, 95^ 59, 76, 69, 68, 26, 35, 44, 78, 89, 199, 97, 76, 89, 98, 94, 57, 69, 78. 6. 59, 58, 67, 68, 49, 98, 95, 57, 49, 58, 57, 68, 69, 79, 87, 88, 279, 368, 599, 594, 29, 37, 48, 49, 57, 198, 197, 179, 939, '9, 39, 46, 42, 57, 68. 7. 235, 267, 319, 429, 487, 576, 387, 563, 397, 876, 875, 587, 939, 948, 952, 269, 268, 479, 89, 97, 88, 89, 99, 57, 58, 89, 76, 378, 492, 865, 296, 93, 39, 29, 36, 76, 78, 88, 89. 8. 76, 49, 48, 47, 46, 57, 59, 56, 86, 58, 89, 89, 28, 29, 37, 49, 39, 65, 66, 79, 89, 47, 93, 58, 57, 79, 76, 77, 89, 98, 99, 197, 58, 67, 69, 84, 56, 189, 95, 78, 97, 79, 89. 9. 78, 45, 69, 78, 64, 28, 39, 49, 58, 97, 98, 78, 87, 88, 96, 69, 67, 74, 76, 88, 89, 98, 276, 398, 892, 89, 74, 59, 96, 87, 88, 96, 98, 96, 88, 92, 76, 75, 89, 289, 274, 98. 10. 472, 477, 88, 99, 79, 86, 57, 58, 96, 279, 375, 49, 58, 97, 78, 88, 38, 56, 9, 8, 19, 18, 29, 36, 47, 58, 98, 98, 376, 49, 389, 98, 97, 365, 69, 37, 47, 59, 48. 12 A MODERN ARITHMETIC 3. SUBTRACTION. Subtraction is the operation of finding how much larger or. smaller one quantity is than another. This may be expressed in two ways : (a) How much must be added to a given number to make another. (ft) By how much must a given number be diminished so as to be equal to another. The quantity from which a second is to be subtracted is the Subtrahend, and the quantity which has to be taken away is the Minuend. The first is usually preceded by the word From, and the latter by Take. The sign ( - ), called minus, indicates the operation of subtraction, and expresses that the quantity following it is to be taken from the quantity preceding it. Thus, 24-13 means that 13 is taken away from 24. The operation of subtraction may be done in two ways, as will be seen from the following example : EXAMPLE. Take 945 from 6128. Th. Arrange as From 6 in addition. Take u. Take 5 units from 8 units, 3 units are left; the 3 is set 5183 down in the units column. Since 4 tens cannot be taken from 2 tens ; take one of the hundreds, and convert it into TO tens; thus making 12 tens in all; subtract 4 tens from 12 tens and we have 8 tens, which is set down under the tens. Next, 9 hundreds cannot be taken from o hundreds ; take one of the thousands and convert it into 10 hundreds, 9 hundreds from 10 hundreds is i hundred which i's set down in the hundreds column. Having taken i thousand from the 6 thousands, 5 thousands remain, which is set down. SUBTRACTION 13 Such is the full reasoning for the operation, but in actual working this is largely curtailed, thus 5 from 8 is 3 : 4 from 2 I cannot : 4 from 12 leaves 8 : 9 from o I cannot : 9 from ten is i. Bring down the 5. Note that practically the subtrahend has been modified thus : From Take Th. The result or answer to a Subtraction Sum is Called the remainder or the difference, and one or other of these words should in the earlier stages of work be always entered. The result can be proved by seeing if when 5183 (the remainder) is added back again to 945 (the number taken away) the first number, 6128, is once more obtained. Thus, 945 Number taken away. ) A , , 5183 Remainder. / 6128 Original number. It can thus be seen if we truly have that number, which, when added to 945, makes 6128. This suggests another method of doing subtraction, known as Complementary Addition. EXAMPLE. Thus, in the same example, 6128 - 945. In the answer let the units digits be represented by U., the tens digits by T., hundreds digits by H., and thousands by Th. 6 i 2 8 is to be the answer when 9451 are added together. Suppose we were adding The Units digit + 5 would equal 8 ; .'. U. = 3. The Tens digit + 4 would make, clearly not 2 but 12 ; .*. T. = 8. I 4 A MODERN ARITHMETIC The i would be carried, and i + 9 + Hundreds digit would equal 1 1 ; .'. H. = i. We should again carry i, i + Thousands digit would make 6 ; .'. TH. = 5. The remainder is .'. 5183. Expressed more briefly, the working of the sum may be as follows : 5+a digit = 8; .'. digit = 3. 4+ =12; :. =8. Carry i. 1 + 9+ =n; /. =i. Carry i. i+ =6; .'. =5. EXERCISES VII. a. Mental or Oral. 1. What must be added to 8 to make (a) 17, (b) 24, (c) 30, (<0 35? 2. What must be taken away from 16 to leave (a) 7, (b) 8, (c) 6, (<05? 3. One boy has 13 marbles in his pocket, a second boy has only 6 ; how many more marbles has the first boy than the second ? 4. A girl has 15 pennies in her pocket, and gives her companion 6 ; how many has she left ? 5. A boy has 30 yards of string, gives some away and finds he has 25 yards left ; what did he give away? 6. A rod is 26 inches long, two pieces are cut off, the first 8 inches in length, the second 5 inches in length ; what is now the length of the rod ? 7. A piece of string is 12 yards long ; what length must be tied on to it, to make it 16 yards long ? 8. What weight must be added to 16 Ibs. to make up a quarter, or 281bs.? SUBTRACTION EXERCISES VII. b. 1. Find the remainders in the following cases, and prove your answer : (a) From 1435 Take 518 (b) From 1063 Take 107 (c) From 1063 Take 914 (d) From 2150 Take 1026 (e) From 839 Take 76 (/) From 2618 Take 1926 (g) From 1764 Take 617 (h) From 8820 Take 312 (*') From looo Take 908 2. (a) 78-25; (d) 309-106; (g) 716-182; ( 3162-1469; (m) 1001 929 ; (b) 100-84 ; (e) 323 - 248 ; (h) 1086-689; (k) 5269-4182; (ri) 1072-974; (c) 262-118 ; (/) 778-90; (0 2175-889; (/) 1000-286; (o) 2116-1119. Supply the missing figures in the following subtraction ex- amples : 3. From 9163 4. From 7642 5. From 1008 6. From 3736 Take Take Take Take 206 3582 4271 7. From 4168 Take 2190 10. From 3333 Take 7865 342i 8. From 7290 Take 6367 11. From 4571 Take 737 9. From 2081 Take .. 12. From 8088 Take 6T62 16 A MODERN ARITHMETIC EXERCISES VII. c. In the following tables, giving the various strata met with in making the wells specified, the depths to the bottoms of the various strata and the well bottom are given ; fill up the table by giving the thicknesses of the different strata, similarly to the worked example. EXAMPLE. Southend Water Works : Surface soil - 3 3 Yellow clay - 33 3 ... by subtracting 3 from 33 Blue - - 417 384 33 417 Running sand - 598 181 417 598 Chalk - - 900 302 598 900 1. Well at New Barnet : 2. Well at Elstree : Strata. Thickness. S-ta. * Tl lickne Yellow clay 24, Yellow clay - 20 Blue III Hard blue clay - 170 JLM\s ,, Quick sand T T C Sandv - 183 Mottled clay 131 kjaunjr ,, ,, i ^fj Sand and water 184 Silver sand- T5C White pebbles - 189 Pebbles in sand n6 Coloured sandy Mottled clav clay - - 196 Dark sand and 37 Clay and pebbles 202 pebbles - 144 Dark sand I C7 Sand and flints - 159 Chalk 439 3. Well at Essendon : 4. Well at Eltham Park : Soil - 2 Strata. lSrt. Th Sand 44 icknes Gravel and sand 26 Yellow clay with Blue clay - 130 fragments of Yellow clay 170 shells - 50 Flints 172 Green sand - 60 Chalk with flints 344 Black pebble bed 70 Sand - - 122 Chalk - - 216 . SUBTRACTION 5. Well at Alperton : Strata. Depth in feet. Yellow clay ' 25 Blue - I6 5 Mottled - 201 Sand - 214 Pebbles - - 218 Flints - 220 Chalk - 400 EXERCISES VII. d. Mental or Oral. EXAMPLE. Find the result of the following operations : 8+ 7 -6+ 9-3-2 + 6 + 8-2-9. 15 9 18 15 13 19 27 25 16 Starting with the +8, the results are figured underneath, as above, the process however being mental ; the answer is 16. Evaluate : 2. 3 + 9+10-5-6 + 8-5 + 4. 4. 8 + 5 + 6-3-4+6-3 + 6. 6. 9 + 2-3-4 + 2 + 4-6 + 8. 8. 2 + 4 + 3-5 + 6-7 + 8 + 2. 1. 5 + 6-4-2 + 7-9 + 6-5. 3. 6 + 7-3-7 + 8+6-9-5. 5. 6-4+9+2-3-6 + 8 + 2. 9. 6-5 + 6-3-2 + 6 + 4-6. 11. 6+4+34-6 + 6 + 4-6. 13. 4+6-3-2 + 6-4 + 3-4- 15. 8 + 7-5 + 6-8-4 + 9-8. 10. 2-1+3 + 6-4-2 + 8 + 2. 12. 8-6 + 3 + 642 + 6 + 3. 14. 3 + 2-4 + 5-2 + 3 + 6-5. 16. 5-4 + 8-5 + 6 + 3-6 + 8. J.M.A. i8 A MODERN ARITHMETIC 4. MULTIPLICATION. Multiplication is only a shortened method of addition, where the addends are all equal. In arithmetical operations the sign for multiplication is represented by x ; if, for example, we wish to find the value of 4 + 4 + 4 + 4 + 4, i.e. of 5 fours added together, the operation is equivalent to multiplying 4 by 5, and is represented by 5 x 4. Similarly 2x8 means 8 + 8. 8x2 means 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2. In 2 x 8 the answer 1 6 is termed the product, the 2 is called the multiplier, and 8 the multiplicand, while 2 and 8 are spoken of as the factors of 1 6. Remembering that multiplication is merely shortened addi- tion, we can construct a set of multiplication tables. 1 two is 2 2 twos make 4, for 2 + 2 = 4. 6 f 3 twos are 2 greater than 2 twos, i.e. 2 I greater than 4. 4 , 82 greater than 6, 9 10 n 10 12 14 16 18 20 22 12 24 8, 10, 12, 14, 1 6, 18, 20, 22. EXERCISES VIII. a. 1. Construct a table of multiples for the numbers 3 to 12 similar to the above, thus 1 three is 3, 2 threes make ... etc. MULTIPLICATION 19 2. Arrange the results of all these tables as in Fig. I (filling in the blank spaces) : where the result of multiplying any number, such as 7 by 8, is put in that square which is under the 7, and in the same line horizontally as the 8. 1 2 3 4 5 6 7 8 9 10 11 12 2 3 4 5 6 7 8 56 9 10 11 12 FIG. i. Note the peculiarity ot the table, that the rows and the columns might be interchanged, for example, and accordingly so also 8x7 = 56 7x8 = 56; 8x7 = 7x8, 6 x 4 = 24 = 4 x 6, This fact may be illustrated graphically on 'squared' or chequered paper. A MODERN ARITHMETIC For example, to show that 4x3 = 3x4. Three squares can represent 3 units, thus (Fig. 2) 3 FIG. 2. four such rectangles will represent the product (4x3) ( Fi g- 3)- But the big rectangle representing the product can be divided up into 3 rectangles each of 4 squares, thus and represents 3x4 (Fig. 4) FIG. 4. To illustrate 3X4. EXERCISES VIII. b. Practical [Apparatus : Ruler and Squared Paper :] Represent graphically the following products, and show in each case that the multiplier and multiplicand can be interchanged : 1. 5x3. 2. 4x5. 3. 6x2. 4. 8x7. 5. 6x9. These results could be generalized by stating that where a and b stand for any numbers. MULTIPLICATION 21 EXERCISES VIII. c. Mental or Oral. EXAMPLE. Give factors for 72. Since 6 x 12 = 72, then 6 and 12 are said to be factors of 72 ; similarly, 8 and 9 are also factors of 72. 1. Give factors for the following numbers : (a) 48, (b) 28, (c) 35, (<*) 90, (*) 88, (/) 32. 2. Write down the results of the following operations : (i) 3x4 + 2; (ii) 5x6-8; (iii) 12x9-9; (iv) 12x9- 12; (v) 8x11+8; (vi) 8x11+3. Multiplication by single digits. The method of multiplying by numbers consisting of a single digit, is best illustrated by considering an example. EXAMPLE. To multiply 876 by 4. Here we have to evaluate 4x876; or, what is the same thing, 876 + 876 + 876 + 876. We do this by addition first. 876 876 876 876 3504 3 2 Next by multiplication : 876 As in addition we added 4 sixes, so we now take 4 4x6 making 24, setting down 4 and carrying the 2 73^7 tens. Corresponding to adding the 4 sevens, and ^ this 2, we obtain mentally the value of 4x7 + 2 or 30, set down o, and carry 3. Next we obtain 4x8 + 3 or 35; and the answer is 3504. 22 A MODERN ARITHMETIC Multiplication by factors. EXAMPLE. To multiply 586 by 45. We have to find the sum of 45 numbers each equal to 586. These numbers could be arranged in 5 groups of 9, for 45 is 5 x 9. The number in each group is 9X 586= 5274, the 5 groups is 5 x 5274 = 26370. The result has been obtained by multiplying, ist by one factor 9, 2nd by the second factor 5 ; />., we multiply in succession by all the factors which make up the multiplier. To multiply by 10, 100, or 1000 is easy, for when multiplied by 10, units become tens, tens become hundreds, and hundreds thousands. Thus, 215 multiplied by 10 becomes 50 + 100 + 2000, i.e. 2150. Notice that the place value of each digit of the original number has been increased tenfold, but that all that has been done is to add a cypher after the unit figure. Similarly, when we multiply by TOO, units become hundreds and tens become thousands ; the place value of each digit is increased one hundredfold, and 2 cyphers are added. Thus, 1 8 multiplied by 100 yields 1800, and so on. EXAMPLE. To multiply 324 by 30. That is, to multiply by a number the factors of which are 10 and 3. Multiplying 324 by 10 we obtain 3240. 3 2 4 > by 3 9720. The two operations are conducted simultaneously, i.e. we multiply by 3 and add a cypher. General Case. EXAMPLE. To multiply 276 by 315. It is required to find the number which would result, if 276 were written down 315 times, and the sum obtained; the MULTIPLICATION 23 result must be the same as if 276 be first taken 300 times, then another 10 times and finally another 5 times. Hence, to obtain the result multiply 276 by 300 first, then multiply 276 by TO, and lastly by 5, and finally add the 3 products together. Or, 315 x 276 = (3oox 276) + (rox 276) + (5 x 276). The working is set down usually as below 276 (i) Multiply 276 by 300, by multiplying by 3 and 315 the addition of 2 cyphers to increase the place ~ value a hundredfold. (ii) Multiply by 10, by adding a cypher to increase the place value of each digit tenfold. I3 * (iii) Multiply by 5. 86940 (iv) Add the three products, and obtain 86940 as the product of 276 and 315. NOTE. It is convenient to place the numbers with their unit digits beneath one another. Generally the work is shortened very slightly by omitting the cyphers used in increasing the place value in the partial products. Thus, 276 828 276 1380 86940 The gain in time is very small, and it is better to omit nothing at first. EXERCISES IX. a. Mental or Oral. 1. A yard (i yd.) contains 3 feet ; how many feet are there in (a) 4 yds., (V) 3 yds., (c) 6 yds., (d) $ yds., (e) 8 yds.? 2. A foot (i ft.) contains 12 inches ; how many inches are there in (a) 8 ft, (b) 6 ft, (c) 4 ft., (d) 5 ft., (e) 9 ft-? 24 A MODERN ARITHMETIC 3. One sixpenny piece has the same value as 6 pennies ; how many pennies should be exchanged for (a) 4 sixpenny pieces, (b) 5, 0) 7, (<t) 2, 0) 8 ? 4. How many days are there in the following number of weeks : (a) 4 weeks, 0) 1 2 weeks, (c) 7 weeks, (d) 9 weeks ? 5. A foot contains 12 inches; how many inches are there in 8 feet? 6. How many marbles are there in 6 bags, each containing 9 marbles ? 7. A yard contains 36 inches ; how many inches are there in a roll of paper 5 yards in length ? 8. A man gives 7 pennies to each child in a school of 160 children ; how many pennies does he give away altogether? Evaluate the following : 9. 25, multiplied by (a) 2, (b) 4, 0) 6, (d) 8. 10. 25, (a) 3, (6) 5, (c) 7- 11. 125, (a) 4, 0)8, 0) 3, (<05, (')? 12. 144, () 3, 0)4, 0)12, 005, 0)6. 13. 236, (a) 2, 0)5, 0) 7, 0)8, 0)9- 14. 121, 0)i i, 0)3, 0) 6, 0)5. 15. 37, (a) 3, 0)6, 0) 9, 0)4- 16. 28, (a) 4, (J)8, 0) 3. 17. 112, (a) 5, 0)8, 0) 9- State all the factors of 18. () 24, (*) 32, 0) 45. EXERCISES IX. b. 1. How many names are there on 9 registers, each containing 45 names ? 2. How many panes of glass are there in 12 windows, each containing 27 panes ? 3. A man has 276 oranges in each of 5 boxes, how many oranges has he altogether ? MULTIPLICATION 25 4. How many lines are there in a page of 9 columns, each of which contains 136 lines? 5. A book has 280 pages with 48 lines to the page ; what is the total number of lines ? 6. What is the weight of 28 blocks of stone, each one of which weighs 137 pounds? 7. If a mile contains 1760 yards, how many yards are there in 26 miles? Multiply the following : 8. 76 by 35. 9. 89 by 47- 10. 57 by 23. 11. 93 by 87. 12. 96 by 64. 13. 296 by 27. 14. 305 by 76. 15. 1905 by 63. 16. 7560 by 640 17. 906 by 267. 18. 693 by 709. 19. 790 by 652. 20. 319 by 1035. 21. 737 by 680. 22. 543 by 706. 23. 31 8 by 7 2 3. 24. 6389 by 38. 25. A store has in it 326 boxes each containing 138 apples, 326 boxes each containing 262 apples ; how many apples are there altogether in the store ? 26. Evaluate : (a) (24 x 260) + (38 x 260) + (37 x 260). 0) (198x325) -(68x325). (c) (3i6xi75)-(3i6x74)- (d} (25 x i4o) + (65 x 140) + (84 x 140) -(170 x 140). (e) (i2X36x25)+(i3oox 14). 27. After working mentally, write down the results of the following operations : (a) 76x5 + 8. (0)37x6+5. (c) 37x6-8. (<*) 124x7+26. (*) 384x9+54. (7)256x12 + 7. (d 382x9+123. W 36x5-4. (2)471x11-8. 00 374x10-32. (k) 256x12-24. (/) 318x7-45. (m) 100-5x6. () 180-3x15. (0)200-11x17. (/) 562 - 7 x 28. (?) 2362-9 x 183. 26 A MODERN ARITHMETIC 28. What is the weight of 312 cart loads of bricks, each one of which weighs 1643 pounds ? 29. What is the distance covered in a year (365 days) by a train making 2 daily journeys of 143 miles ? 30. If a pint of a liquid weighs 23 ounces, how many ounces short of looo are there in a cask containing 43 pints ? 31. A boy wishes to join up by a thread, two posts 4956 inches apart ; he has 12 skeins of thread each 415 inches in length ; how much thread has he over ? EXERCISES IX. c. Mental or Oral. Write down the product of the given numbers, and the numbers corresponding to the digits marked with an asterisk. Thus in 1. (a) 234x146. The result would be 234x40=9360. 1. (a) 234 by 146 ; (b) by 164 ; (c) by 2046 ; (d) by 713. 2. (a) 893 by 363 ; (ff) by 363 ; (c) by 3001. Write down the products of the pairs of numbers represented by digits marked by asterisks in the following : 3. (a) 876 by 325 ; (b) 876 by 523 ; (c) 8672 by 253. (d) 9613 by 2*182 ; (e) 7992 by 899 ; (/) 8734 by 734- Squared and Cubed Numbers. When a number is multiplied by itself it is said to be squared, thus 12x12= 144, and 144 is the square of 12 MULTIPLICATION 27 The operation is generally represented by the index 2 written above the number, thus 144= 12 X I2=I2 2 16 = 4 x 4 = = 3 x 3 = = 1232 = a 2 , where a stands for any number. 9 = 3 x 3 = 3 2 I23X 123 = 1232 Similarly, a 3 = a x a x a a 4 = axaxaxa 3 4 = 3 X 3 X 3 X 3 = 81 ; an d hence 81 is called the 4th power of 3. Notice that three like numbers have to be multiplied together to obtain the third power ; four to obtain the fourth power, and so on. EXERCISES X. 1. Write down (a) the powers of 2 up to the loth, (b) the powers of 3, also up to the loth. 2. Write down the powers of 5 up to the 5th. 3. Name the 5 lowest powers of 2 which are also powers of (a) 8, and of (b) 16. 4. Construct a table of squares for the first 20 numbers. 5. Using your table of squares, find the numbers which, when squared, will give the numbers nearest to the following : (a) 10, (V) 20, (c) 30, (d) 40, (e) 50, (/) 60, (g) 80, (h) 1 10, (0 130, (/) 150. 6. Construct a table of the cubes of the first 20 numbers. 7. A number is known to be (a) a squared number, find the possible digits in the units place ; what digits are possible if the number is known to be (b) a cubed number ? 8. Write down as concisely as possible : (a) a x a x a x a x a, (b) b x b x b x b x b x b, (c} (d) y xy, (e) zxzxzxzxzxzxzxz. 2S A MODERN ARITHMETIC FIG. 5. To represent the square of 5. EXERCISES XI. Practical. [Apparatus: Squared Paper, ,] 1. The accompanying figure represents the square of 5. Draw similar figures representing the results of squaring the following numbers : 3, 4, 8, 11, 13, 21. 2. Show by drawing figures that 3x4 + 3x5 is the same as 3x9. 5x2 + 5x3 5x5, II X 10+11 X I II 2 , 14x10+14x4 i4 2 . 3. Show by drawing figures that (a) 3 x 13 is the same as 3 x io + 3 2 , () 4x14 4Xio + 4 2 , (c) 3x23 3x20+32, (d} 20X23 20 2 +20X3, (e) 6x36 6X30 + 6 2 , (/) 30x36 3o 2 + 6x3o. 4. Draw a square to represent 36 2 , divide it up into two rectangles one of which represents 6 x 36, the other 30 x 36. Divide the rectangle representing 6 x 36 as in (e} in question (3). 30x36 (/) (3). Show from your divided figures that 3 6 2 = 3o 2 + 6 2 + twice 30 x 6. 5. Draw figures showing that (a) i3 2 =io' 2 + 3 2 + twice 3X 10, (b) i8 2 =io 2 +8 2 + 8xio, (c) 24 2 = 20 2 +4 2 + 4x20. 6. Making use of the method suggested by (5), write down the squares of the following numbers : 00 J 5> 25, 35, 45, 55 5 0*) '7, 22, 33, 91, 82 ; (c) 102, 203, 305. MULTIPLICATION 29 7. Draw as in the accompanying diagram a figure to represent the square of 10 ; divide it up into 10 parts, as in the figure, and write down the num- bers corresponding to each part ; thus the shaded portion in the diagram is the 8th part and represents 15, the first represents I, the second 3 and so on. (a) Find the sum of the first 3 parts, (*) ,, 4 and so on up to (c) the first 10 8. Draw a figure to represent the following sum : FIG. 6. and find the sum. 9. Represent any odd number, say (a) 33, as the difference between two squared numbers ; also (b) 29, (c) 31. 10. A man puts 9 pennies, resting against one another, with their centres in a straight line, against these he puts another row of 7, then another row of 5, and so on ; how many rows will there be altogether, and how many pennies ? 11. A man puts 24 rows of shillings against each other, i in the first row, 3 in the second row, 5 in the third row and so on ; how many shillings has he ? 5. DIVISION. Division is the operation of ascertaining how often one quantity is contained in another, or how many times the same quantity may be taken away from another. From this second aspect of the operation, division may be looked upon as repeated subtractions where the minuend remains constant. Thus, 7 is contained in 50, 7 times with a remainder of i ; or, 7 may be subtracted 7 times from 50 and i remains. Hence (i) a concrete quantity may be divided by a concrete 30 A MODERN ARITHMETIC quantity of the same kind, as, how often is 7 contained in 21 ? (2) A concrete quantity may be divided by an abstract one, as, divide 12 cwts. into 4 equal parts. (3) An abstract quantity may be divided by another abstract quantity, as, divide 28 by 4. The quantity to be divided is called the Dividend; the quantity we divide by is known as the Divisor, the result obtained as the Quotient ; and when a quantity is left at the end of the operation it is called the Remainder. It is evident that Divisor x Quotient + Remainder = Dividend. The explanation of the process is seen from the following example : EXAMPLE. It is required to divide 3419 by 7. Arrange the work thus : ij\ 3419 Since 3 is less than 7, regard 3 thousand , and 4 hundred as 34 hundred ; divide 1 r< by 7 ; the result is 4 hundred as quotient and 6 hundred as remainder ; V 7x4 + 6 = 34. Regard this 6 hundred remainder and the i ten as 61 tens ; divide by 7 ; the result is 8 tens as quotient and 5 tens as remainder ; V 7x8 + 5 = 61. Regard this 5 tens and the 9 units as 59 units ; divide by 7 ; the result is 8 units as quotient and 3 as remainder ; V 7x8 + 3 = 59. The answer or quotient is 488, remainder 3. To prove the result, see if Divisor x Quotient + Remainder = Dividend. Thus (7 x 488) + 3 = 3416 + 3 = 3419 = Dividend. The operation of division is often represented by the symbol +, which signifies that the quantity before the sign is DIVISION 31 to be divided by the quantity following it. It is also expressed in other ways; thus 12 divided by 4 may be written as -/-, or 12/4. EXERCISES XII. a. Mental or Oral. 1. How often is 8 contained in 48, 80, 144 and 192? 2. 9 63,117,288,72? 3. How often can 11 be taken away from 100, 200, 154, 362 ? 4. A basket containing 124 apples is shared equally among some boys, giving each one 8. How many boys will have shares? What apples, if any, remain ? 5. What must 12 be multiplied by to give 84, 120, 48 and 480 ? EXERCISES XII. b. 1. Divide 2375 by (a) 3, (b) 5, (c) 7 ; prove your result in each case. 2. What number when divided by 7 yields 285 as quotient and 6 as remainder ? 3. How many numbers less than 1200 are multiples of 8 ? 4. There are 12 inches in i foot ; how many feet are there in 3002 inches ; how many inches over ? 5. A flagon can hold 7 pints ; how often can it be filled from a cask containing 1250 pints? 6. How many sixpences are equivalent to 3276 pennies? 7. A man has 2561 pennies ; what is the least number of pennies he must add to them, so that he can exchange them for an exact number of shillings ? Division by Factors. When a divisor is the product of 2 or more factors, the division may be performed by each of these successively. The 32 A MODERN ARITHMETIC final result is the quotient. A little care is required in determining the remainder when the factors do not divide the quantities exactly. EXAMPLE. To divide 7563 by 32. It is required to find how many lots of 32 can be taken from 7563, and how many units will then remain. 4175^3 8 1800 remdr 7 ( Le ' 756s can be divided into l8 9 S rou P s of 4, and I V &\ 3 units over. 236 remdr. 2 (i.e. 1890 groups of 4 are equivalent to 236 groups \ of 32, and 2 groups of 4 over. Quotient = 236 with remdr. 2x4 + 3, or n units. Hence total remainder is 2 groups of 4, + 3, or 1 1 units. The same factors may have their order changed, thus : 4 945 remdr. 3. 236 remdr. i. Quotient 236 with i group of 8 and 3 units, or n units, as remainder. This method can be extended to 3 or more factors. EXERCISES XII. c. Perform the following operations, using factors when advisable. In each case, when factors are used, change their order and thus verify your own work. 1. 2694 - (a) 4, (b) 5, (c) 6, (d) 7, (e) 8, (/) 9, (g) u, (h) 12. 2. 28407- (a) 14, (b) is, (c) 1 6, (d) 18, (e) 21, (/) 24. 3. 30269- (a) 27, (b) 32, (c) 35, (d) 36, (e) 40, (/) 44. 4. 1 125 - (a) 45, (b) 48, (c) 49, (d) 55, GO 60, (/) 64. 5. 2784 - (a) 72, (6) 84, (c) 96, (d) 105, (e) no, (/) 112. 6. 3269 - (a) 135, (b) U7, (c) 1 68, (d) 180, (*) 192, (/) 210. DIVISION 33 EXERCISES XII. d. Graphic. [Apparatus : Squared Paper.] 1. Divide graphically 82 by (a) 7, by (b) 17, by (c) 13. (Mark off a distance representing 82, and find the number of times a distance representing 7 can be marked off on this.) 2. Mark off a distance representing 97, and see how many distances representing 5 can be marked off on this, next take these distances in groups of 3 ; how many groups are there, what is left over ? 3. Similarly, divide 86 by (a) 15 (i.e. by 3x5) and by (b) 24. 4. The accompanying figure contains 1 66 small squares. Divide it up so as to show that when 166 is divided by 28, by the method of factors, first by 7 then by 4, the remainder is of the form 3x7 + 5, and equal to 26. What is the Quotient ? FIG. 7. 5. Draw in a similar manner to that employed in (4) a figure divided up to show the form of the remainders when 166 is divided by 35) by 42 and by 63. 6. Draw figures as in (4) and (5), showing the forms of the remainder when 154 is divided by 72, by 54 and by 45. J.M.A. 34 A MODERN ARITHMETIC Division by the General Method. EXAMPLE : 41 184 -r 234. 6 7 o 411 hundreds odd is greater xoo than 100 times 234 and less 234 ) 41 184 ( than 200 times 234. 23400= 100 times 234 We subtract therefore 100 times 234 from 41184, and the X 77M . remainder is 17784; and we 16380 = 70 times 234 set I00 above 4Il84 as in the 1404 example. 1404 = 6 times 234 17784 we know is less than 100 times 234; we have to find the greatest exact ten multiple of 234 less than 17784; now 2 hundred and odd divides into 177 hundred odd, clearly something nearly 80 times, and by multiplication 80 times is found to be too large ; 70 times 234 is however less than 17784, and we accordingly multiply 234 by 70 and subtract from 17784, leaving 1404; we set the 70 above the 100. By inspection it is seen that 234 divides into 1404, some- thing like 6 times ; 6 is tried, and found to equal 1404 exactly, and the result of the division is 176. The method is usually contracted as follows : 1 76 = Quotient 234141184 234 1778 1638 1404 1404 To prove the result, test if Quotient x Divisor + Remainder = Dividend. Here 234X 176 + = 35200 + 5280 + 704 = 41184. The result is therefore correct. DIVISION 35 Italian Method. The writing work is sometimes abbreviated by writing down the remainders only in the division. EXAMPLE : To evaluate 87643 + 387. Worked out in full. 226 = Quotient 387 87643 774 1024 774 2322 181 = Remainder. Worked out by Italian method. 2 26 = Quotient 387 (87643 twice 7 = 14, 14 and 2 = 16, carry i. 1 1024 8= 16, with i = 17, 17 and 0= 17, carry i. 2 53 3=6, with 1 = 7, and 1 = 8, so obtaining 181 the ist partial remainder 102. Bring down the 4. twice 7 = 14 + 0=14, carry i. 8= 16, with i = 17, 17 + 6 = 22, carry 2. 55 3=6, with 2 = 8, 8 + 2=io, so obtaining the 2nd partial remainder 250. Bring down the 3. 6 times 7 = 42, and 1 =43, carry 4. 6 ,, 8 = 48 with 4 = 52, and 8 = 60, carry 6. 6 3= 18 with 6 = 24, and 1 = 25, and the final remainder is 181. 36 A MODERN ARITHMETIC Division true to the Nearest Unit. Frequently in division we are not concerned with the remainder, and it is more convenient to state the answer true to the nearest unit (or to the nearest ten, or hundred). EXAMPLE. 87643 -f- 387. The quotient is (as on p. 35) 226 true to the nearest unit, as 181 is less than one half of 387. Again, in dividing 7836 by 9 the quotient is 870 and 6 over, but stated true to the nearest unit the quotient is 871, as the remainder 6 is greater than one half of 9, and 9x871, i.e. 7839 is nearer to 7836 than 9 x 870 or 7830. EXERCISES XII. e. Find the quotient and remainder when each of the following numbers (1)62341, (ii) 56375, (iii) 88216, (iv) 70062, (v) 34562, (vi) 16324, (vii)83iio, are divided by each of the following : (0)23, (3)56, (^123, (*94?8, (*) 525, (/) 417, (g) 73i, (^) 922. EXERCISES XII. f. Mental or Oral. 1. What is the least number which can be taken from 60 so that the remainder may be divisible by (a) 8, (b) 7, (<:) 9 ? 2. What is the least number which can be added to 73 so that the sum may be an exact multiple of (a) 6, (b) 7, (c) n, (d) 5? DIVISION 37 3. In division operations, supply the missing quantities in the following : Divisor. Quotient. Remainder. Dividend. (a) 8 3 4 * (*) 9 * * 232 M 7 123 5 * (d) 6 182 3 * ( e ) * 7 4 200 (/) * 12 10 43- 4. What digit must any number end in, to be exactly divisible by 5? 5. What are the numbers less than 100 exactly divisible by 25 ? 6. What must be the tens and units digits of any number if it is exactly divisible by 25 ? 7. State the remainders when the following numbers are divided by 25 : 80, 180, 237, 257, 983, 933, 1953, 201, 441. 8. Multiplying by 100 and dividing by 4 is equivalent to multiplying by 25. Making use of this, state a quick way of multiplying by 25. 9. Evaluate the following : 25x126, 25x18, 25x144, 25x48, 25x337. 10. State a method of quickly dividing by 25. 11. Evaluate 23oo-r25, 375-^25, 875-1-25,- 1925-5-25, 47oo-r25. 6. FACTORS. Factors. Many rules can be established for finding possible factors, thus : (a) 2. Clearly all even numbers are multiples of 2. () 4. Since 100 is 25x4, any number which is an exact multiple of 100 is an exact multiple of 4; therefore any \ 3 8 A MODERN ARITHMETIC number divisible by 4 must have the number represented by the tens and units digits also divisible by 4. Thus, 1872 is divisible by 4, since 1800 is divisible by and since 72 is also divisible. To test if 2316 is divisible by 4, we merely note that 16 i divisible by 4. (<:) 8. Similarly, since 1000 = 125 x 8, any number is divisible by 8 if the number represented by the hundreds, tens and units digits is divisible by 8. (d) 9. To test if a number is divisible by 9. Consider any number such as 3426. 3426 = 3000 + 400 + 20 + 6 = 3 x(iooo) + 4x(ioo) + 2 x (io) + 6 = (3x999 + 3) + (4x99 + 4) + (2x9 + 2) + 6. But 999 is exactly divisible by 9, also 99 and 9 The number 3426 can be divided by 9 if 3 + 4 + 2 + 6, i.e. the sum of the digits, can be divided by 9. And the remainder, when 3426 is divided by 9, is the same as that obtained by dividing 3 + 4 + 2 + 6 by 9. EXERCISES Xin. 1. Show that 1836, 2176, 3024, 51232, 6328 are exactly divisible by 4. Which are divisible by 8 also and which by 9 ? 2. Find the remainders when the following operations are gone through : (a) The difference between 3265 and 2356, divided by 9, and also by 4. (b) The sum of 3265 and 2656, divided by 9, and also by 4 and by 25. (c) Twice 7361 divided by 9. 3 and FACTORS 39 3. If a cubic foot of iron weighs 450 pounds, how many cubic feet can be obtained from 26732 pounds ; what weight will be left over? 4. The distance between London and Sheffield is 162 miles ; during which consecutive journey will a locomotive journeying backwards and forwards between these two places complete loooo miles ? 5. A number has two of its digits unknown, ia8$& ; they are represented by a and b. The remainder when the number is divided by 25 is known to be 6, and known to be 3 if the divisor is 9 ; what are the two digits ? 6. In (5) find the possible numbers, if both are known to be exactly divisible by 36 (i.e. by 4 and by 9). 7. Supply the missing figures in the following example on division, worked by the Italian method : 3** = Quotient 237 ) ***** 1251 668 1 94 = Remainder. 8. How many pieces of cloth 128 inches in length can be cut off from a roll 3646 inches long ; what length will be left over? 9. How many numbers less than 1500 are exactly divisible by 51? 10. A number is divided by another by the method of factors, thus : a \ ***** )**** rem. 3 *** rem. 4 Total remainder = 35. What is the first divisor a ? A MODERN ARITHMETIC 7. ARITHMETICAL AVERAGES. Arithmetical averages. The word average is used in a variety of senses ; but when the sum of a number of quantities is divided by that number, the result is termed the arithmetical average of the given quantities. EXAMPLE. Find the arithmetical average number of M.P.'s on the different sides in the Parliaments quoted below : Year. Ministerial Side. Opposition Side. 1833 480 Whigs 173 Conservatives 1835 380 Whigs 273 Conservatives 1837 352 Whigs 301 Conservatives 1841 367 Conservatives 286 Whigs 1847 327 Whigs 326 Conservatives 1852 333 Liberals 320 Conservatives 1857 366 Liberals 287 Conservatives I8 59 348 Liberals 305 Conservatives Here there are 8 ministries. Adding the numbers] Total on Total on . on the Ministerial and [ Ministerial Side, Opposition Side, Opposition Sides. 2953. 2271. Dividing each total by the number of ministries the arith- metical average is obtained. .*. 2953-7-8 = 369 is the average on the Ministerial Side to the nearest member, and 2271 -i-8 = 284 is the average on the Opposition Side to the nearest member. EXAMPLE. The average number of merchant and other vessels (not including warships] launched in the United Kingdom from 1893-1903 inclusive is given in the table below. Find the arithmetical average of the number launched per year : ARITHMETICAL AVERAGES Year. Number of Ships. 1893 536 1894 614 I8 9 5 579 I8 9 6 696 1897 59i I8Q8 761 I8 99 726 19OO 692 I9OI 639 1902 694 1903 697 Add the number of ships and divide by the number of years (n) to obtain the arithmetical average. Total number of ships = 7225. Arithmetical average = 657 to the nearest exact number of ships. EXAMPLE. During the four years 1902-1905 inclusive, the arithmetical average of the number of ships (not warships] under construction in the United Kingdom was 412 per year, the number during 1902, 1903, 1904 were 388, 393, 393 respectively. How many ships were under construction during 1905 ? Since the average number per year for 4 years = 41 2, the total number during the 4 years = 1648, the total number during 1902-04 = 388 + 393 + 393 .'. number during 1905 = 1648- 1174 = 474- H74; EXERCISES XIV. N.B. In all the examples below, the average asked for is the arithmetical average, and the answer is to be found to the nearest unit. 1. There are 10 bags of oranges, the bags containing the following numbers of oranges : 8, 9, 1 1, 7, 6, 9, 7, 9, 8, 7. Find the average number of oranges contained in each bag. 2. There are 5 piles of coins, 10 coins in one pile, 8 in the next and 6 each in the remaining three. Find the average number of coins in each pile. 42 A MODERN ARITHMETIC 3. In a school there are 13 boys in Form I., 12 in Form II., 14 in Form III., 15 in Form IV., 10 in Form V. and 6 in Form VI. Find the average number of boys in each Form. 4. The number of depositors in the Post Office Savings Bank for the years 1894-1903 inclusive is given in the accompanying table. Find the average number of depositors (in thousands) during that time : Number of Depositors in Year. Post Office Savings Bank. In thousands. 1894 6,109 I8 95 6,454 1896 6,862 1897 7,240 1898 7,631 I8 99 8,047 I9OO 8,440 I9OI 8,788 1902 9,i33 1903 9,404 5. (a} Find the average number of letters received per head per annum at the various Post Offices in the United Kingdom during the period as below : Year. Letters Per Head. 1896-1897 48 1897-1898 5 1898-1899 54 1899-1900 55 I9OO-I9OI 56 I9OI-I9O2 59 1902-1903 60 1903-1904 61 () Draw up also a table showing the yearly increase in the number of letters per head, and calculate the average yearly increase. ARITHMETICAL AVERAGES 43 6. Find the average number of seamen in the British Navy during the period -1889- 1890 to 1904-1905. Year. Seamen. 1889-1890 51,526 1890-1891 54,918 1891-1892 56,995 1892-1893 49,133 1893-1894 51,428 1894-1895 57,026 1895-1896 61,945 1896-1897 65,757 1897-1898 70,472 1898-1899 75,709 I899-IOOO 70,322 1900-1901 82,821 I90I-I902 85,323 T902-I003 88,691 1903-1904 93,385 1904-1005 101,287 7. Find the average number of day schools inspected (a) England and Wales, (b) Scotland, (c) Ireland from 1894-1903 inclusive. () ENGLAND AND WALES. (b) SCOTLAND. (c) IRELAND. Year. No. of Schools No. of Schools No. of Schools Inspected. Inspected. Inspected. 1894 19,709 3,053 8,505 1895 19,739 3,34 8,557 1896 19,848 3,083 8,606 1897 19,958 3,086 8,631 1898 19,937 3,067 8,651 I8 99 20,064 3,062 8,670 I9OO 20,100 3,104 8,684 1901 20,116 314! 8,692 1902 20,153 3,M5 8,712 1903 20,173 3,149 8,720 44 A MODERN ARITHMETIC 8. The average mean temperatures at Greenwich, in degrees Fahrenheit for each month excepting November, of a certain year are: January 37, February 39, March 41, April 45, May 53, June 59, July 62, August 61, September 55, October 50, December 40. If the average temperature for the 12 months is known to be 49, find the average temperature during November. 9. The average dates at which the following spring migrants arrive and leave the British Isles is given below : Cuckoo April 14 Aug. 25. Swallow April 10 Oct. 15. Treepipit April 15 Sept. 25. Sand-martin April 9 Oct. 4. Martin April 14 Oct. 12. Nightingale May 8 July 27. Landrail April 10 Oct. 5. Find the average dates of arriving and of leaving of the birds as a class. 10. Find the average (a) population, (b) acreage, (c) number of emigrants from the counties of Ulster at the times specified : COUNTIES. Population. 0) Extent in Acres. (f) Irish Emi- grants. ULSTER. 1901. 1891. 1903. Antrim and Belfast co. boro 461,240 711,276 3,042 Armagh - 125,238 313,036 758 Cavan ... 97,368 467,025 978 Donegal ... 173,625 1,190,268 1,045 Down 289.335 611,927 Fermanagh 65,243 417,665 310 Londonderry and co. boro 144,329 513,388 855 Monaghan - 74,505 318,806 403 Tyrone ... 150,468 778,943 962 MISCELLANEOUS EXERCISES 45 EXERCISES XV. Miscellaneous. Graphic. [Apparatus : Squared Paper .] 1. Draw a figure such as ABCD, representing the square of 17 ; also AEFG, representing the square of 13. Produce GF to H, mark off D C K CK equal HB and complete as in the figure. Show from your figure that 2. that I?"- 13 -3X4- Draw figures showing i8 2 -7 2 =25x ii. i6 2 -io 2 = 26x6. 3 o 2 - 292 = 59x1. 1 H E B FIG. 8. State in words the general result you have obtained. 3. In the accompanying figure, note the numbers represented by the figures marked a, t>, c ... . Then use your figure to find the sum of 1+2 + 3 + 4+5+6 + 7 + 8 + 9. 4. Draw a similar figure to find the sum of (a) i, 2, 3, 4, 5, 6, 7, 8. (d) i, 2, 3, 4, 5, 6, 7. 5. Find, by a similar method, the sum of (a) I+3 + 5 + 7 + 9+ 11 - i h A r / c </ c b a i 2 3 4 5 6 7 8 9 FIG. 9. (b) 1+4 + 7+10+13+16+19. (<:) i + 3+ 5 + etc., until there are 7 terms. (d) 1+4 + 7+ io+etc., until there are 8 terms. (<?) 1+2 + 3 + 4..., until there are 12 terms. C/) 3+5 + 7 + "-> until there are 7 terms. 46 A MODERN ARITHMETIC EXERCISES XVI. Miscellaneous. 1. In four villages the populations of which are 2952. 896, 1528 and 962 ; 3267 are males. What is the total population and how many females are there ? 2. Three heaps of bricks containing 7860, 9428 and 12064 are to be cleared away in cartloads of 345 bricks each. How many loads will have to be taken and how many bricks left ? 3. The population of a town on the'ist Jan., 1904, was 78,563. During the year 942 fresh people came to live in it ; 329 people left it ; there were 1964 births and 1428 deaths. What was the population at the end of the year ? 4. A vessel has to take a cargo of 3470 tons. Of this 1874 tons are coal, 1473 tons coke, and the remainder is cement. What is the amount of cement ? 5. At a football match there were 134 people on each of 13 rows in the grand stand ; 7 times the number in the second seats ; and in the third seats 476 more than in the other two places. What was the total number of spectators ? 6. What number must be added to 7342 to make it equal to the product of 879 by 19 ? 7. The restoration of a church requires ^6500. Towards this, there are 3 subscribers of ^100, 14 of ,50, 32 of ^25, 8 of ^20, 70 of ,10, 75 of 5. What amount is unsubscribed for? 8. A house has 4 floors, with 8 windows to each floor, and each window has 12 panes. How many panes would 32 such houses have? 9. What is the least number which must be added to 718456 so as to contain 85 an exact number of times ? MISCELLANEOUS EXERCISES 47 10. By how much must 5384 be increased so as to be equal to 10,000 ? 11. In a long division sum the quotient was 7054, the divisor was 572 and the remainder 342 ; what was the dividend? 12. To the sum of 9832 and 5716, add their difference. 13. Among how many pensioners may ^2100 be divided so as to give each one .37, and what sum remains undivided ? 14. Find the sum of the prime numbers between 60 and 100. N.B. A prime number is one which has no factor other than unity. 15. The water from a coal pit is being pumped up by an engine at the rate of 3 strokes a minute, and delivering 350 gallons at each stroke. What amount is pumped in 24 hours ? 16. The product of two factors is 1580877 and one of the factors 15783. What is the other ? 17. Multiply one million one thousand and one by one hundred thousand one hundred, and express the result in words. 18. The quotient in a division is 479, the dividend is 3476418 and the remainder is 794 ; what is the divisor ? 19. What is the smallest number which, when subtracted from 99099, will make it exactly divisible by 909 ? 20. When 128 apples are taken away 67 times from a store there are 44 left. How many were there at first ? 21. From 98,765 subtract any other smaller number consisting 01 the same five digits in a different order, and divide the remainder by 9. Why must the remainder be divisible by 9 ? 22. Multiply together 1,634,797 and 73,946, and explain any short method you know for verifying the correctness of the product. How is it adapted for the verification of a quotient ? 4 s A MODERN ARITHMETIC 23. The following is a summary showing the value of the exports from the United Kingdom of articles manufactured (or partly so), excluding foods, drinks, tobacco, and ships. Find the totals. DESTINATIONS. () 1893. w i8 94 . to I**- N <*) 1896. Certain Foreign Countries : Germany 14,853,000 14,550,000 17,297,000 19,175,000 Belgium 5,429,000 6,829,000 6,516,000 6,947,000 Holland 8,628,000 8,137,000 6,725,000 7,542,000 France - 10,293,000 10,135,000 10,649,000 11,034,000 Russia - 5,295,000 5,632,000 5,807,000 6,023,000 Italy - 3,216,000 3,044,000 3,437,000 3,476,000 United States 21,087,000 16,464,000 24,985,000 18,129,000 Total of above) Countries J 24. Values are given of Imports into, and Exports from, Tunis in each of the years 1892 to 1899 inclusive, distinguishing separately trade with France, Inter-Colonial trade, and trade with Foreign Countries. Add up the quantities horizontally, filling up the entries under the heads of Total. IMPORTS. Years. From France. From other Colonies. From Foreign Countries. Total. i ,000 Francs. i ,000 Francs. 1,000 Francs. i,ooo Francs. 1892 20,774 2,665 15,887 1893 21,726 I,6l3 15,044 1894 22,942 i,955 17,026 I8 95 23,309 1,382 19,395 1896 25,563 i,536 19,345 1897 27,872 2,537 23,411 1898 29,876 2,073 21,572 I8 99 34,264 1,851 19,663 MISCELLANEOUS EXERCISES 49 Years. EXPORTS. To France. To other Colonies. To Foreign Countries. Total. I8 9 2 1893 1894 1895 1896 1897 1898 I8 99 1,000 Francs. 20,965 14,521 19,874 26,348 20,223 22,179 24,992 26,714 1,000 Francs. 6,105 4,892 6,038 4,579 5>796 4,078 5,i37 4,245 1,000 Francs. 10,130 10,272 11,021 10,320 8,488 10,474 14,068 18,474 T,OOO Francs. 25. Values are given of (a) Imports into, and (b) Exports from, Algeria in each of the years 1892 to 1901 inclusive, distinguishing separately trade with France, Inter-Colonial trade, and trade with Foreign Countries. Add up the quantities horizontally, filling up the entries under the heads of Total. IMPORTS. Years. From France. From other Colonies. From Foreign Countries. Total. i,ooo Francs. 1,000 Francs. i,ooo Francs. 1,000 Francs. 1892 189,639 6,114 57,323 1893 184,754 4,110 50,834 1894 199,319 n,345 54,470 1895 203,163 6,563 51,486 I8 9 6 217,802 5,326 52,671 1897 216,175 3,889 56,837 1898 225,535 6,742 69.946 1899 26o,422 3,979 55,447 1900 259,355 5,47i 58,992 1901 258,977 6,596 70,545 J.M.A. A MODERN ARITHMETIC Years. EXPORTS. To France. To other Colonies. To Foreign Countries. Total. 1892 1893 1894 1895 1896 1897 I8 9 8 1899 I9OO I9OI 1,000 Francs. 200,191 148,415 213,848 253,293 203,780 245,979 232,137 279,675 173,467 207,099 1,000 Francs. 3,177 3,373 4,170 9,848 6,785 7,3 r 3 8,892 11,966 12,329 7,492 1,000 Francs. 40,599 40,840 47,695 44,646 36,845 42,436 44,740 54,774 56,521 52,250 1,000 Francs. 26. Find the totals in the following table : Statement showing the Tonnage of Foreign Sailing Vessels, distinguishing the Nationalities, entered with Cargoes at Ports of the United Kingdom from the under-mentioned British Colonial and other Possessions during the year 1902. () 0) fc) (rf) 60 (/) Norwegian. German. Italian. Russian. Swedish. Other. Channel Islands 654 97 831 Burma - 1,926 2,722 Australia West Australia - South Australia - New South Wales 32,257 2,648 1,808 11,677 7,128 5,637 2,760 2,529 3,683 3,023 1,568 2,547 1,460 1,657 4,830 3,309 Victoria 3,318 3,284 3,780 1,711 3,302 Queensland ... 1,243 New Zealand 2,934 i,339 3,359 800 I,l85 Canada Newfoundland 156,998 4,54i 3,466 IO,IOO 779 4,231 5,208 2,88 5 357 West Indies - British Honduras - 3,468 6,849 1,437 491 870 ... 3,054 British Guiana 2,788 1,464 All other 3,900 ... 758 1,115 Total - MISCELLANEOUS EXERCISES 51 27. (a) The destination of the coke exported by the Westphalian Coke Co. has been as under : Find the total export each year. DESTINATION OF COKE EXPORTS. 1900. 1901. 1902. Metric Tons.* Metric Tons. Metric Tons. France .... 784,688 782,220 710,870 Belgium .... 188,689 66,216 153,947 Holland .... 40,052 41,195 47,778 Switzerland ... 95,055 103,030 100,305 Italy 24,805 35,338 33,687 Spain - - - - - 16,255 5^750 30,327 Austria 302,634 237,422 186,405 Russia 88,158 42,793 75,612 Sweden -,-'- 33 ? 523 29,315 32,665 Norway - 13,392 12,780 21,378 Denmark - 21,023 13,913 15,167 England - 850 1,540 2,346 Roumania - 2,950 Greece - 3,820 4,120 3,375 Asia Minor - 1,630 Asia 4,385 6,047 5,807 Africa - 502 1,300 1,990 America - ... 39,268 71,112 183,895 Australia 2,487 5.000 1,845 Total * A metric ton, or Tonneau equals roughly 2200 lb., that is, about 40 Ibs. less than the British ton. (b) Find also the average yearly exports during the period 1900-1902, to (b) Austria, (c) Denmark, (d} America, (e) Asia, Africa and Australia. 5- A MODERN ARITHMETIC VALUE OF SOME EXPORTS TO GERMANY FROM THE UNITED KINGDOM OF ARTICLES MANUFACTURED IN THE UNITED KINGDOM. ARTICLES. 1899. 1900. 1901. Tj | rvMiifo/1 *7 T *7OT 60 Q7 I 72 WQ Caoutchouc ..... / 1 5/ U1 205,934 u y>w i 179,792 /^ovy 118,023 Chemical products .... 289,975 306,248 253. 10 7 Coal, products of .... 242,674 292,907 238,251 Cotton : Yarn ..... 1,894,124 1,762,873 1,573,250 Manufactures - - - - - 1,770,261 2,081,044 2,144,402 Cycles 32,868 18,567 20,250 Earthen and china ware 134,5" 106,673 86,008 Glass 39.599 47.524 38,499 Grease, tallow 105,995 72,655 76,412 Haberdashery and millinery 42,327 39,749 35,9oi Hardware and Cutlery 154.475 139,024 126,630 Hats of all sorts 51.038 40,3*9 45,000 Implements - 79,060 52,792 48,383 Leather ...... 409,122 413.776 389,823 Linen : Yarn ..... 213,932 223,360 192,227 Manufactures 281,042 287,547 247, 102 Machinery and Millwork : Sewing machines - ... 148,774 178,803 150,526 Steam engines 324,182 3H,7i7 213,128 All other sorts 1,715.367 1,547,277 1,174,282 Manure ...... 329,769 256,089 386,710 Medicines, Drugs .... 56,459 53,044 6 5.5 2 3 Metals, Iron and Steel : Tin plates 326,910 276,710 148,930 Other iron and steel 2,427,121 2,645,283 1,100,755 Copper - 491,176 350,443 435,299 All other metals .... 327,779 353,673 301,641 Oil and floor cloth .... 56,653 42,686 27,262 Oils Painters' Colours .... 140,727 129,646 229,546 125,539 255,298 110,836 Paper of all sorts .... Silk: 58,134 58,742 52,085 Thrown, twist or yarn 77,594 52,476 27,260 Manufactures 84,136 77,372 79,281 Stationery, other than paper 73.453 63,411 67,412 Wool, flocks, noils, waste, and combed Woollen and worsted yarn - Yarn, alpaca, mohair, and other sorts - Woollen and worsted manufactures 1,035,673 3,159,112 1,168,625 976,327 652,102 2,863,939 1,022,259 879,903 637,605 2,i4i,559 1,086,721 848,697 BRITISH MONEY SYSTEMS 53 (Use the values of Exports given in the Table on page 52 for Exercises 28, 29 and 30.) 28. Find the total value of the following Exports in the three years 1899, 1900, 1901 : (a) Cotton yarn and manufactures. (b) Haberdashery and millinery and hats. (c) Linen yarn and manufactures. (ei) Silk of all kinds. (e) Wool, and woollen and worsted yarn. (f) Machinery and millwork. (g) Metals of all kinds. 29. Find the total value of Exports in (a) 1899, (fy I 9> (c) 1901. 30. Find the average value of the Exports for the three years under the heads of (a) cotton, (b) linen, (c) cotton and linen together, (d) machinery and millwork. 8. BRITISH MONEY SYSTEMS. THE coins used in the payment of British money, with their relative values, are as follows : 4 farthings = 2 halfpennies = i penny. These are the only " copper coins ; " they are styled copper, although actually made of an alloy. 1 2 pence = i shilling. 20 shillings = i sovereign or pound sterling. A number of pennies is expressed by writing a 'd' after the number, a halfpenny by J</., a farthing by -]-</., and three farthings by \d. Shillings are expressed by numbers with s. after them, and sovereigns by numbers with after or before them. Thus, ^3. 14^. 8f*/. expresses three pounds, fourteen shillings, eight pence and three farthings. 54 A MODERN ARITHMETIC In terms of the pound, shilling, pence and farthing all sums of money are expressed, but, for convenience of exchange, use is also made of the following coins : Threepenny Piece worth 3 pence Sixpenny Piece 6 (6d.\ Florin Half Crown Fourshilling Piece 2 shillings (25-. or 2/-). 2 shillings and 6 pence (2*. 6d. or 2/6). 4 shillings (45-. or 4/-). 5 (3 s - or 10 (IQS. or io/-). Crown Half Sovereign A gold coin called a Guinea, having the value of 2U., was formerly current in Great Britain, but is not now used, though payments to professional men are frequently expressed in guineas. It is useful to be able to express sums of money either in pence, or in shillings, or in pounds, and vice versa. EXAMPLE i. Express 2. 35-. $d. in pence. 2. $s. 5^ = 435-. $d. Y 2 are equivalent to 2 x 20 shillings; 435. $d. = $2id. Y 43-r- are equivalent to 12 x 43, i.e. 516 pence; or shortly, 2. $s. 5^ = 43^. $d. = $2id. EXAMPLE 2. Express 2349 farthings in pounds, shillings and pence. 2349 farthings = 587^. Y 4 farthings = id. 587^ pence =485. ii\d. Divide 587 by 12, since i2d. = is. 485-. 1 1 \d. = 2. 8.9. 1 1 \d. Divide 48 by 20, since 2os. = i t or briefly, 2349 farthings = 587 \d. = 485-. 1 1 \d. = 2. 8.T. 1 1 \d. EXEECISES XVII, a. 1. Draw up a table showing the value in pence of each number of shillings from I to 20. 2. Draw up a table showing the value in shillings and pence of 20 pence, 30 pence, 40 pence, ... up to 100 pence. BRITISH MONEY SYSTEMS 55 EXERCISES XVII. b. Mental or Oral. 1. Give the pence values of : (a) 2s. 6J, (b) y. */., (c) ^ 2cL, (d) 5 j., (e) 6s., (/) 7 s., (g} Ss. : ... tO 2OS. 2. State the value in . s. d. of: (a) 300 pence, (b) 240^., (c) 500^., (d) 6ood., (i) 720^., (/) 840^., Or) 9o^j (^) 960^., (z) loocW. EXERCISES XVII. c. 1. Express the following sums in their pence equivalents : (a) 3. 14* 7^-, (^) ^7- i8j. 6rtl, (^) ^49. i6j. 2</., (^) ^59- H-y- ii^-, (e) ^704- oj. 5^., (/) ^9674- i3^- i^-, Cr) ^7005. i2j. 4^., (/z) ^17023. i8j. Id. 2. How many sixpences are there in the following sums of money ? (<0 73-, W ^14- 7J- 6rf., (?) 203. iy. (*?., (^) ^7053- i i-f- 6flT., (^) ^2963. oj. 6^., (/) ^1934. 2s. 6d., (g} ^18347- u. 6^., (//) ^10304. i9j. 6^/. 3. Reduce each of the following sums to threepences : (a) 17. sj. 9^., (^) ^29. 6j. 3^., (r) ^294. yj. 6^/., (rf) ^4834- <w. 9^., (^) ^200. oj. 3^0 (/) ,965- i8j. 6^., G?) 23045- i-f- 9^- (*) ^597- iQJ. 4. Convert the following sums into their equivalent number of guineas : {a) 21, (b) 31. ioj., (r) ^47. 5J., (^) ^95. iu. } (^) ^154. 7^. ; and the following into pounds and shillings : (/) 135 guineas, (g) 738 guineas, (h) 537 hall-guineas, (/) 3149 half-guineas. 5. Express each of the following amounts in pounds, shillings and pence : (a) 759 farthings, (b) 3159 farthings, (c) 6289 half-pennies, (d) 3576 pence, (*) 19053^-, CO 7I93 6 farthings, Or) 793 threepenny pieces, (h) 9452 sixpences. 5 6 A MODERN ARITHMETIC 9. COMPOUND ADDITION. EXAMPLE. Find the sum of 27. &s. <)d., ,31. 125. ioj^., ^157. os. $\d. The amounts are arranged in vertical columns, keeping pounds under pounds, shillings under shillings, etc. j~ , The operation is as follows : (a) Obtain the total of the farthings = 5 9 , farthings = i \d. ; set down the \d. under the I0 | farthings and carry the id. to the pence T 57 _ 5% column. ^216 2 ij () Similarly, the total of the pence ^~~~^~~~ ~ ~~"~ ' column with the id. added = 25^. = 25. id.; the id. is set down under the pence column and 2S. carried to the shillings column. (c) The total of the shillings column with the 2S. carried = 225. =1. 2S.; the 2s. is set down under the shillings and the i. is carried to the pounds column. (d) The pounds are summed as in simple addition, so that the total is ^216. 25. i\d. The sums to be added may be arranged horizontally and then added, care being taken not to confound the figures of the different names. Thus the sum of . Ss. <, i. izs. 10., 1. os. ^. = 2i6. 25. id. EXERCISES XVIII. a. Mental or Oral. Find the totals of the following : 1. (a) 4K &*, 7K 5^-5 (6) "d., aj., iod., 6d; (c) $&., 6frf. 9^., 2\d. ; (d) 8^., is. 3^., gd., $d., iod., $d. ; (*) u. 4^., 2j. 6^. 2d., 4s. ; (/) yj. &/., 3-y. 4^., 9^. 2^., i8j. 6^. 2. (^) 135-. 5*/., 2J. 5^"., 2^.; (^) 8s. $d., 6s. yd., 2s. 2*/., 2s. iod. (c) is. 4\d. t is. 2^., is. o%d., io^d. : (d} js. 6d., 3^. 6d., is. &d., $s. $d. (e) 2s. yi., 4J. 8^., 7^., 6s. ; (/) 10^., is. 8^., 5^., 4.9. COMPOUND ADDITION 57 EXERCISES XVIII. b. Find the sum of the following amounts : 1. ^29. i6s. $d., 38. 15-$-. 3<, ^102. os. $d., 93. iSs. 4^., y. 7 Id. 2. ^304. 15*. id., 796. iSs. 7$d, 42. los. $d., 1903. 175. $d., 3. ,750. i8j. &, ^3059. 17* 10^., ^9. 7.y. 3 </., ^845. 6s. gd., ^7364. iij. 9^. 4. ^384. oj. 8^., ^7132. 9^ "i^-, ^34- i5 J - I0 ^j ^4i3- i5- r - 8^., ^70569. 14.9. ;</. 5. ^76. 13^- 4^d., ^903. i8j. 9|^., ^5760. 5j. 8|^., ^9732. 14^- 7^-, ^376. 19^. 5^., ^ioo 10. 15.$-. gd. 6- ^3045- I7J- lo//., ^94. y. 5^., ^897. i w., ^7045- 6j. 3^., ^903. isj. 8^., ^6904. i 3 j. 8|^. 7. ^693. 4s. l\d., ^843. i6j. o\d., ^9763. or. 3^., ^584. 18.9. gd., ^3692. 4j. 3^., ^619. i2j. 7^, ^48- 7J- 3<*> ^7043. 13^- 9^ ^3. 8s. 7 <t., ig. i8s. $d, 12s. &d., 39- 6f. 8^. 8. 7. i&r. 9^., ^5. I7J. 3^., ^409- 13^. oi^- 5 ^72. i&. S$d., j. 7\d., ^2094. I5J. 8^., 13. i4j. 5J//., ^783. 12^. iij/f., 9. What is the total of the following collection : 7 sovereigns, 13 half-sovereigns, 1 1 crowns, 29 half-crowns, 37 florins, 93 shillings, 231 sixpences and 853 pennies? 10. Find the total value of 84,762 penny newspapers and 105,763 half- penny ones. 11. At a concert the following amounts were taken : 193 half- crowns, 213 florins, 419 shillings and 97 sixpences. What were the total takings ? 12. The takings at a shop were: on Monday 17. 8s. $d., on Tuesday ^13. 13.?. 9^., on Wednesday ^i. 3^. io</. more than on Tuesday, on Thursday 20. us. 6d.. on Friday ^8. 13^. 4^., on Saturday 2. js. \o^d. more than on Thursday and Friday together. What was the total amount ? 5 8 A MODERN ARITHMETIC 13. What is the money value of 93 sixpenny magazines, 5093 penny papers and 7153 half-penny papers. 14. The moneys paid every fortnight as wages by a certain manufacturer in 3 months were: 79. ly. &/., ,68. 15.?. 4^., ^82. 19^. 5</., 76. I2s. 7</., 77. 15-r- 9<* and %i. 15^. 9</. What was the total amount, and how many sixpences would it be equal to ? 15. At a certain turnstile the following payments were received : 1951 sixpences, 5035 threepences and 1751 pennies. What was the total ? 16. 5 persons A, B, C, D, E form a pool : A puts in 500, B 496 shillings more than A ; C puts in as much as A and B ; D puts in 473 guineas and 980 crowns ; E as much as C and D. What is the value of the pool ? 10. COMPOUND SUBTRACTION. The process of subtracting one amount from another of like kind may be seen by considering an example. EXAMPLE. Take 9$. 14^. $\d. from 200. 9*. Place the larger amount first with the smailer under it. . s. d. Proceed as follows : (a) \d. cannot be taken 200 9 3i from \d. Take id. from the $d. in the 93 J 4 5! pence column. \d. from a id. is J*/., which with the \d. makes d. This is set down I4 9* under the farthings. (b) $d. cannot be taken from zd. Hence take is. from the gs. ; and $d. from is. is 7^., which with the zd remaining in the pence column gives yd. This is set down in the pence column. (<r) 14.9. cannot be taken from 8s. \ is taken from the column. Now 145. from a leaves 65-., which with the Ss. remaining gives 145. This is set in the shillings column. (d) Taking ^93 from ^"199 leaves ;io6. (e) The remainder is ;io6. 145. COMPOUND SUBTRACTION 59 EXEECISES XIX. a. Mental or Oral. 1. Take 8f<tf. (a) from is., (b) from u. 6^., (<:) from 2s. 6d., (d} from 4s. id., (e) from 7^. 6d., (/) from IQJ. 9^. 2. Take io|^. () from 2s., (b} from 2J. 6^., (c) from 10^., (^/) from 15,9. 2 t -/., (*?) from 17.9. 6d., (/) from 2U. 3. How much is is. gd. (a) short of 4^., (b} short of js. 6d., (c) short of ios. ? (d) short of i$s. &d., (e) short of 20^.? 4. What must be added to 13^. ^d. (a) to make 18.?., (^) to make ^i, (c} to make /I. i^.? 5. What must be taken from 20.?. (a) to leave 14^. 1\d.^ (b) to leave us. iod., (c) to leave 75. j^d., (d) to leave is. if*? EXERCISES XIX. b. Find the difference between the following amounts 1. (*) i7. 5-r- 8 ^- and ^9- 3-f- 10^. (^) ^20 and ^i i. oj. $\d. (c) ^34. I05-. 9^. and ^36. 3^. $d. (d) $i. i8s. io\d. and ^32. 16^. nd. 2. (a) ^2091. 13^. jd. and ^184. oj. iod. OJ - and s - 6d- and (d} ^59630. i8j-. 5^. and ,70159. os. id. 3. (a) ,13509. ly. 2%d. and ^684. 15^. \\d. (b) ^10149. 5.9. 6d. and ^994- 17-f- 8 ^- (*) ^5^39- HJ. 7^. and ,50119. i2s. & (d) ^70436. os. *$d. and ^6945. iu. 6d. 4. () ^20563. i;j. *>\d. and ^18137. 18^. 9! (b} 75632. 15^ 4^ and ^7085. 3^. 10^. ^480,060. 13^. 4^. and ^8119. los. 8d. 60 A MODERN ARITHMETIC 5' ( rt ) , I0 9>5 10 - os. gd. and 965 1. os. \o\d. (b) ,311,045. i6s. lod. and ,732. 14*- ll k d - (c) ,94537. os. 2d. and ,27608. i8s. -$\d. (d) ,200,163. 5-y. 6;/. and .9037. 12.?. Sd. 6. How much larger is .752. 13^. jd. than .692. 15.?. 8^.? 7. In order to pay a bill of ,20. 15^. gd., how many sixpences in addition to ;i8. 17^. $d. will be required? 8. By how much must ,345. 4s. $d. be increased so as to become ,400 ? 9. A house was sold for .920. 1 5-r., which a year previous was bought for 873 guineas. How much was the gain or loss ? 10. The takings at a fete consisted of 2074 pennies, 927 three- pences, 2215 sixpences, 1195 shillings, 513 half-pennies. What number of half-pennies would make the total sum taken ,150? 11. A lady took ,2 to market, and spent the following sums in cash purchases : meat 8s. $\d. t butter 5^. 3^., eggs is. A^d., bacon 2s. 3^., sundry vegetables 45-. $^d. What sum remained ? 12. A clerk in giving change for a ,5 note gave 3 half-sovereigns, 19 half-crowns, twelve shillings, 9 sixpences, and the rest in three- penny pieces ; how many of these had to be given ? 13. A debt of ^3500 was contracted. Two payments of ^493. 18^. jd. and 2839. i6.y. 8d. respectively were made. What sum remained unpaid? 14. The receipts of a Company during the first 4 weeks of the year 1905 were ,1736- i$s- 9^-> ,843. 14^. 7^., .2019. 6s. $d. and ^1875. 19^-. 10^., while the receipts in the next 4 weeks were ^1839. us. 6d., ,951. 14-$-. id., 2016. $j. gd. and .1528. 13^. gd. Find the difference between the takings of the first month and those of the second month. 15. The correct amount of a bill to be paid was .19. los. 5^., but in a hurry it was given as ^19. $s. io$d. What was the amount of the error, and in whose favour, the payer's or the person who received the money ? COMPOUND MULTIPLICATION 61 11. COMPOUND MULTIPLICATION. The procedure may be seen by considering an example. EXAMPLE. What is the value of 9 copper bars, each worth 2..1$S. IOJ</. /-.:. j ( a ) 9 halfpennies = 4^., the \d. is set &' s ' ' down under the farthings column and the 2 *5 I0 2 qd. carried on to the pence column. (b) 9 x io*/. + 4d. = 94^. = ]s. lod. -, the 25 2 loj io</. is set in the pence column and the 75. carried on. (c) 9X 15*. + 7*. = 1425. = 7. 2s.; the 2s. is placed under the shillings column and ^7 carried on. (d) 9x^2 + ; 7 = ^25. The result, or the amount, is ^25. 2s. io\d. When the multiplier is a composite number, that is, one which can be split into factors, the multiplication can be performed by multiplying successively by the factors. For instance, multiplying by 36 might be done by 6 times 6 4 times 9, or 12 times 3. Thus, ;n. ios. %\d. multiplied by 36 = ;ii. ios. 8J^. x 12 x 3 = ^138. Ss. If the multiplier be a non-composite number and large, or if the factors are either unknown or inconvenient, the method of splitting up the number into a number of tens and units, or hundreds, tens and units, etc., is often adopted. A method however, of multiplying each term of the sum of money by the total multiplier and converting the products into the next equivalent or higher term may be used. Indeed it may often be expedient to adopt both, so that the pupil may check his accuracy. Thus, let it be required to multiply ^27. 13,?. 8J^. by 319. The sum may be done by process (A) or (B). 62 A MODERN ARITHMETIC EXAMPLE (A). To multiply 27. 13*. 8J</. by 319. * d. 27 13 8^x9 10 276 1 6 10 J = 10 times the amount 10 2768 89= TOO 3_ 8305 6 3 =30 276 16 ioj= 10 249 3 2\= 9 8831 6 3f = 3i9 >, EXAMPLE (B). 319 times 2-]. 135-. = 6 7i &/. x ,, = 2552^ = 212^.8^.= 10 12 8 135. x = 4147*. = ............ 207 7 o 8613 _ 8831 6 3} Ofttimes various devices will suggest themselves to the ingenious and thoughtful pupil in dealing with multiplication exercises. e.g. 319 times ^27. 135. %\d. 320 times ^27. 135. 8J^. -^27. 13*. 8x 10-^27. 13 EXERCISES XX. a. Mental or Oral. Find the cost ot 1. 7 articles at 6^/., 8*/., 9!^., is. 2d., is. ?>\d., $s. 8^., $s. . 9^., 8.y. i \\d. each. 2. 8 articles at 3f*/., 5^., 8|//., 9^., u. 3^, 2J. 5^., 3^. n . 7^., 13^. 7^. each. COMPOUND MULTIPLICATION 63 3. 12 articles at i\d., 3^., 4%d., 5^., 8|</., 9f</., u. i^/., is. ?>\d., 19^. iod?., 2cw. 4|^. each. Note that in dealing with dozens each penny in the cost of a single article corresponds to a shilling in the cost of a dozen. Articles i^/., 2dl, $d. ... each are u., 2s. y. a dozen, and articles \d., \d., \d. each are 3^., 6</., 9^/., that is, quarter of a shilling, half a shilling or three quarters of a shilling a dozen. Use this principle in the exercises marked with a star. 4. 13 articles at 2|^., 8^., 9^., io^/., u. 3^., u. 8^/., 2^. 3^., 35-. 5^., 3J-. n^*., Sj 1 . icW., 9J-. iifrt'*. 5. 6 articles at 4^. jd., $j. 3^., 8j-. 7^., 9.?. 8^., 9^. EXERCISES XX. b. 1. Multiply ^73. 9-y. \o\d. by 6, 8, 10, 12. 2. ^358. os. u%d. by 7, 9, 11, 5. 3. ^49- ifo- 8|^. by 16, 13, 19, 14. 4. ^795- 19* lod - b y l8 ' I2 I 5> ! 7- 5. ^843. 14.$-. 7|rf. by 11, 10, 15, 19. 6. ,, ^6095. I2.y. 6\d. by 24, 36, 48, 60, checking your results by using 2 sets of factors for each. 7. ,i93- 15* 4i<* by 72, 80, 84, 128, 125. 8. ^273. Zs. 9|^. by 192, 240, 256, 105. 9. ^597- 15^- *&- by 303, 78i, 634, 999. 10. ^738. i&r. 9f^. by 792, 543> 729, 1024. EXERCISES XX. c. Problems. 1. What is the total cost of 39 tons of coal at 8^. 9^. a ton and 43 tons at gs. 3<? 2. What are the total wages of 8 men at 5^-. 3^., 23 women at 2s. ^d. and 9 boys at is. iod.? 3. What will be paid for a flock of sheep numbering 583 at i. iqs. d. each? 64 A MODERN ARITHMETIC 4. Find the rent to be paid for 137 acres of land at 2. is. yd. per acre. 5. What will be the cost of 309 tons of ore at js. $$d. per ton ? 6. The cost of a ton of coal at the pit's mouth is 3^. A,\d. per ton, the railway carriage from the colliery to a ship is $s. 6d. per ton, the dock charges and tipping 2\d., the freight $s. 2d. and the carriage from the ship to the final destination is. 6d. What will be the final cost of a cargo of 1950 tons ? 7. A man purchased 73 head of cattle at ^3. 6s. gd. each and their keep cost .21. los. 6d. What did they cost the purchaser altogether ? 8. 1 8 bricklayers working 6 days a week, of 9 hours per day, receive lo^d. each per hour. What will their wages amount to in a month ? 9. Find the cost of 365 yards of steel rails weighing 66 Ibs. per yard, at $d. per Ib. 10. What sum is required to pay the wages of 137 men each earning 2. 14^. jd., 58 women each earning i. 2s. Sd. and 35 boys at 19^. ^d. each ? 11. The rent charge on a farm is the value of 246 bushels of wheat at 4s. jd. per bushel and half the number of bushels of barley at 2s. gd., in addition to a fixed sum of ^26. 145-. ^d. What is the yearly rental ? 12. What is the value of 63 railway shares at ^143. 8s. gd. each? 13. What sum is necessary to purchase 37 houses at ^285. 1 3.?. gd. each ? 14. How much cheaper is it to buy 73 score of articles at Ss. gd. per score, than at $^d. each ? 15. The amount of wages paid at a colliery every fortnight is ^546. 13.9. Sd. What is that per year'?. 16. What is the value of the yield of 3 fields with a total acreage of 138 acres, yielding on an average 2 tons of hay per acre at 4. is. gd. per ton ? 17. What is the cost of gas at an establishment where 54,000 c. ft. are used for lighting and 29,000 c. ft. for cooking and heating, the former being 4^. jd. and the latter 3^. $d. per 1000 cubic feet? COMPOUND MULTIPLICATION 65 18. An artizan's weekly earnings being taken as ^2. 14^. gd. per week and his expenses as ^29. i&s. yd. per quarter, what does he save in a year ? 19. What receipts would be obtained from 29 first-class passengers at 2. $s. lod, each, 78 second-class passengers at i. gs. gd. and 135 third-class ones at iSs. $d. each ? 20. What will be the value of 18 barrels of oil, averaging 44 gallons each at i i^d. per gallon ? 12. COMPOUND DIVISION. The examples in division may be of two kinds : either (1) to divide a certain sum of money into a given number of equal parts ; or (2) to find what number of sums of one value will amount to a sum of some higher value. EXAMPLE. The sum of 79. Ss. \\d. was divided equally among 9 persons ; what did each person receive ? Arrange either as (A) or (B). 9 1 79 8 i| ^79. 8s. i&. + 9 = ;8. i6s. 8 16 51 Proceed as under : (a) 9 divided into ^79 (as described in simple division) gives ;8 with 7 over undivided : the ^8 is set down under the in (A) or placed to right as in (B). (b) The ^7 over together with the 8^. = (7 x 2os. + 8^.) = 148.$-., and 148^. divided by 9 gives quotient 165-. and 4^. over undivided : the i6s. is placed in its proper column or position. (t) The 4.?. over together with id. = (4 x \zd. + id.} = 4<)d., and 49^. divided by 9 gives 5^. and ^d. over : the $d. is set in its proper column or position. J.M.A. E 66 A MODERN ARITHMETIC (d) The 4</. over with the ^. = (4 x 4+ 2)= 18 farthings, and this divided by 9 gives 2 farthings or \d. y which is set down at the end. The quotient, or share of each person, is 8. i6s. $\d. If the divisor is greater than 12, but can be split up into convenient factors, it will be frequently convenient to use them. EXAMPLE. To divide ^138. los. by 48. Factors of 48 are (a) 4 and 1 2, or (b) 6 and 8 ; using (a) we divide first by 4 and then by 12. -L f /Y x^o * 138 10 o 12 34 12 217 8J Ans. The following is an example of the general process of compound long division : EXAMPLE. What sum of money could be given to each of 357 persons from ^3742. 35. iod.? What would remain over 1 } . s. d. . s. d. 357)3742 3 10(10 9 yf 172 (a) Dividing 357 into ^"3742, we 20 have 10 and ^172 over. 3443 shillings. W *1 2 . = 344*-, adding on the 3-y. we obtain 34435-. 12 W 3443-r- divided by 357 yields 9.y. and 230^. over. 2770 pei ^ 230^. = 2760^., adding on the 2 ' r " i ok 2770^. is obtained. _! (e) 2770^. divided by 357 yields 1084 farthings. 7^. and 271^. over. 13 (/) 27 1//. = 1084 farthings. 1084 farthings divided by 357 yield 3 farthings and 13 farthings over. .'. A payment of ;io. 9$. *i\d. could be made to each person, and 13 farthings or $\d. would remain over. COMPOUND DIVISION 67 EXAMPLE. How many payments of 2. 6s. g\d. can be made with 2%. los. $d.? It is obviously necessary to reduce each quantity into its least common term, which is here farthings. The work can then be arranged as follows : 2. 6s. <)\d.= 465-. <)\d. = $6i\d. = 2245 farthings. 28. us. $d. =571^. $d. = 6855^. = 27420 farthings. By simple division 27420-1-2245 = 12 and 480 over; there fore 12 payments of 2. 6s. q\d. can be made, and the sum of 480 farthings or 105. will remain over. EXERCISES XXI. a. Mental or Oral. 1. Divide 4s. 8d. by 4, 7, 8 and 16 ; and 6s. 8d. by 4, 5, 8, 10, 40. 2. 7s. 6d. by 9, 30 ; and 8s. gd. by 5, 7, 8 and 21. 3. I2s. 6d. by 4, 5, 6, 8, 15 ; and 18^. 4d. by 4, 5, 1 1, 22. 4. 2. 8s. 4d. by 5, 8, 16 ; and 2. 14*. gd. by 4, 6, 9. 5. 13. 7s. 6d. by 4, 5, 6, 7, 8, 9, 12, and note the amount of undivided money remaining in each case. 6. How often is $%d. contained in 2s. 4^., 2s. lid., 8s. 2d,, us. id. and 20.?. 5</.? 7. How often may payments of Sd. each be made from 3^., 5^., 6s., 8s. 4d., los. 6d., i2s. What sums remain over or undivided in each case ? 8. Divide 19. 14.?. yd. by 5, 7, 9, n. 9. ^216. 19^. 4d. by 6, 8, 10, 12. EXERCISES XXI. b. 1. Divide ^693. 15* by 1 6, 1 8, 21, 24. 2. 7352. 9*. *d. by 32, 45, 52, 56. 3. 29418. i.v. M. by 81, 96, 105, 112. Note in each case the value of the undivided amount at the end of the operation. 68 A MODERN ARITHMETIC 4. Divide ^2032. 13^. &d. by 53. 5. ,6614. 1 5* 4^ by 79- 6. ,17831. i6-r. nf^.bySs. 7. ,108163. i2s. $\d. by 247. 8. ,346827. i6j. %\d. by 439- 9. 6160706. us. lod.by 764. 10. ;i 56143. los. by 427, and give the value of the undivided sum remaining. 11. Divide ,25194. i6s. 2\d. by 97, and give the value of the remainder. 12. How often must the sum 01 5^. gd. be put aside to save ;i9. 1 6s. 9</.? 13. In what time will a weekly saving of 3-y. 7\d. amount to 75. &.? 14. Divide ^138 between 27 men and 36 boys, giving a man twice as much as a boy. 15. What is a man's average daily earnings when his yearly income is ;i75- 13^. i\d.1 16. Prize money to the amount of ^2787. los. 6d. was divided among a company of sailors, giving. each one ,35. 14^. 9^. How many were there ? 17. What is the greatest number of sums of 2. i&s. ^d. which can be formed from a donation of ,100 ; and what amount remains ? 18. The average cost per child of giving a number of children a week's stay in the country is gs. 8^., and a sum of 50 guineas has been collected for this purpose. How many can be taken and what sum remains unused ? 19. A collection containing an equal number of half-crowns, florins, shillings, sixpences and pennies amounted to 14. $s. lid. What was the total number of coins ? 20. How many rupees, each worth is. iofy/., would be equivalent to the sum of ,286. 6^. $d. ? COMPOUND DIVISION 69 21. How many quarters of wheat, each worth 53.9. &/., together with the same number of quarters of barley each worth 32^. 6d.. would be equivalent to a rent of ,249. ijs. 8^.? 22. Complete the missing terms in the given table of income or wages, stating the missing terms to the nearest farthing. (The yearly wage is given, and from it has to be found the monthly wage, weekly wage, etc., taking 12 months to the year, 52 weeks to the year and 365 days to the year.) Per Year. Per Month. Per Week. Per Day. (") . s. d. I O O . s. d. o i 8 . s. d. . s. d. h\ CQO (r> IO O o 16 8 (rf) So o c (A IOO O O (f\ 500 o o (V) IOOO O O \/ EXERCISES XXII. Miscellaneous Problems. 1. The daily takings at a shop in i week were ^14. ly. $d. t 7. i8j. 6</., 10. 7s. 9%d, iS. 15*. 5^., 12. 4*. gd. and ^35. gs. 6d. What was the arithmetical average daily receipt ? 2. 14 barrels of oil were bought for ,25. Ss., 17 barrels more for .39. 8.y. 5^. and 26 others for ^33. IDS. id. If they had been bought at a uniform price per barrel, what would that price have been? 3. How many French francs, each worth 9^/., would be equivalent to 95 Brazilian milreis of 2s. -$d. each ? 4. What sum remains out of a yearly income of ^1526, after a weekly expenditure throughout the year of 2,6. gs. $d.t 5. From the yearly receipts of a company, amounting to ,79,020. 4-y. lod. the sum of 7$fi37- S s - 4^- ' s P a ^d for working 70 A MODERN ARITHMETIC expenses, a sum of 2000 is set aside to form a reserve fund and the remainder divided equally among the 47 shareholders. What does each shareholder receive ? 6. If I exchange 1 120 Ibs. of sugar at i\d. a Ib. for an equivalent value of rice at $d. a Ib., how many pounds of rice shall I receive ? 7. At a certain hotel in the winter season of 16 weeks, 15 fires each burn on an average i ton of coal in 80 days. What is the value of the coal used at 25^. gd. per ton ? 8. A collection realized 343 silver coins : viz. crowns, double florins, half-crowns, florins, shillings, sixpences and threepenny pieces, and there was an equal number of each. What was the value of the collection ? 9. A sum of ,25,000 is to be raised by the issue of 650 first- class shares of 27. I2s. 6d. each and 1127 second-class shares. What is the value of a second-class share ? 10. At a manufactory, a weekly levy of 2d. in each received in wages is made toward the doctor's fund. There are 128 hands employed, half earning ,2. 6^. $d. a week, and the other half i. I2s. 6d. each. What does the levy amount to in a year ? 11. The sum of .1950 ship prize money is to be shared first among 9 officers, each to have ,175, and the remainder among the ship's crew, giving each 7. \6s. -$d. What was the total company of the ship ? 12. I wish to convert 42 Bank shares at .33. 6^. $d. each into 13 Railway shares at .113. 8s. qd. each. What additional cash sum will be necessary ? 13. To defray a debt of ,1500, 64 sheep worth 2. 3^. f)d. a piece, 36 cows each worth ^15. i6s. yd. and 17 horses each worth ,34. i2s. 6d. are sold. What sum is still owing? 14. I exchange 742 yards of silk, at gs. 3\d. per yard, for a cash sum of ,79. i6s. id. and a number of yards of cloth at 4s. yd. a yard. What number of yards of cloth must be given ? 15. The holiday expenses of a party of 45 were ,108. 3^. yi.\ 1 8 paid 2. \2s. 6d. each, 16 paid 2. gs. 6d. What had each of the remainder to pay ? MISCELLANEOUS PROBLEMS 16. A cargo of 2344 tons of English coal is sold in San Francisco at i. 6s. <)d. per ton What is the nearest number of dollars, each worth 4s. 2d., that will discharge the amount ? 17. Butter is bought at 6 guineas for 112 Ibs., and sold at is. 4^d. per Ib. What quantity must be sold to realize a total profit of 10 guineas? 18. By selling 57 yards of cloth at 2s. ^d. a yard, a loss of i was made ; while later in the season 73 yards of the same cloth were sold at 2s. lo^d. per yard, making a profit of i. 14s. io^d. What was the buying price of the whole per yard ? 19. Which is cheaper, and by what amount, to buy a 9-ton truck of coal at i8j. gd. per ton, with haulage 2s. 6d. per ton extra, or to buy the coal in sacks delivered at 2s. 6d. each, 10 sacks making up a ton ? 20. A cart load of apples has 640 Ibs. of apples of prime quality which are sold at $d. a Ib., and 760 Ibs. of second quality at 3f</. What difference would there be in the receipts if all were sold at $d. a Ib. ? 21. Find, correct to the nearest penny, the average rate of interest per cent, declared by the following Life Assurance Companies : Name of Office. Rate of Interest per cent. . s. d. Provident - 3 15 o Provident Clerks - 3 H 3 Prudential (Ordinary) - 368 Refuge (Ordinary) 380 Rock 3 17 o Royal 3 13 9 Royal Exchange - 3 16 i Sceptre - 3 16 2 Scottish Accident 3 IS i Scottish Amicable 3 i? 9 Scottish Equitable 410 Scottish Imperial (Ordinary) 3 16 o A MODERN ARITHMETIC 22. Find, true to the nearest penny, the average Imports and Exports per head per annum in the United Kingdom from 1892- 1903 inclusive. Year. Imports Per Head. Exports Per Head. . *. d. . s. d. 1893 10 10 I 5 13 4 1894 10 9 ii 5 ii i 1895 10 12 3 5 15 2 1896 II 2 II 6 I 2 I8 9 7 ii 5 4 5 i? o 1898 II 12 10 5 15 6 I8 99 ii 17 9 697 1900 12 14 2 7 i 6 1901 12 II 3 6 14 9 1902 12 II 10 6 15 i 1903 12 l6 I 6 17 3 23. What was the average price per head of cattle, of which 60 were bought for ,10. i8.r. 6d. each, 60 for ,10. i6s. <)d. each, 50 for 10. 15^-. 3*/. each and 30 for 10. 12s. $d. each ? 24. Add together ^175. 17^. 6</., five hundred guineas, eighty- seven half-crowns and 1143 fourpenny pieces. 25. Find the whole cost of 20 dozen boxes of fruit at 14.?. J\d. per box ; 40 dozen boxes at 13^. 9^. and 60 dozen boxes at 12^. M. 26. Two horses (Bob and Farmer) and a carriage cost ,250. Bob and the carriage cost ^181. 13.9. yd., while Farmer and the carriage cost ^155. 6s. t>d. What was the separate price of each of the three? 27. At a collection for a Sanatorium there were 79 sovereigns, 135 half-sovereigns, 69 crowns, twice as many half-crowns, 30 score and 6 shillings, 1145 sixpences and 593 pennies. What was the total amount collected, and what was it short of ,500 ? 28. From 271 times ^35. 4*. 2</. take ,9441. 6^. M., and divide the remainder by 89. 29. A boy bought a knife for 2s. 6d., and afterwards sold it for gd. and 210 marbles worth a penny a dozen. What did he. lose by the sale? MISCELLANEOUS PROBLEMS 73 30. A flock of 54 sheep is sold at 2. $s. 6d. each and a second flock of 78 is sold at 2. 7s. gd. each. What was the average price per sheep to the nearest penny ? 31. The average cost of a sheep in a flock of 648 was 2. i^s. gd. ; 420 of the sheep were worth a total of ;iooo. What was the average cost of the remaining sheep to the nearest penny ? 32. A farmer having 240 acres pays a rent of .2. los. per acre on 100 acres, of .3. $s. per acre on 80 acres and ,4. $s. ^d. per acre for the remainder. What is the average rent per acre ? 33. If 30 tons of coal at 2os. 6d. a ton be mixed with 22 tons at 24.?. per ton, find the cost per ton of the mixture, true to the nearest penny. 34. A man borrowed ,200, of which he paid at Lady Day ,40. los. yd., at Midsummer ,68. los. 8d. and at Michaelmas 2$. us. id. How much of the ^200 does he still owe, and how many half-crowns will pay it ? 35. The sum of ,17 3s made up of the same number of sovereigns, half-sovereigns, half-crowns, shillings and sixpences. How many are there of each? 36. How many payments of ^74. i6s. $d. can be made with ,5000, and what sum is left? 37. How many payments of ,86. iSs. gd. each can be made with .1000, and what sum is left? 38. Among how many men can ,5828 be divided, giving each man ,1. i8.y. gd., and what money, if any, is left? 39. In a bag of 2760 coins there is an equal number of sovereigns, half-sovereigns, crowns, half-crowns, florins, shillings, sixpences and threepenny pieces. What is the number of each kind of coin, and their total value ? 40. If the total revenue of a country is ,107,780,179 and the population is 34,908,560, what is the average amount of revenue raised per head of the population (true to the nearest penny) ? 41. A prize of ,6374. iSs. 6d. was divided between 9 officers, each receiving ,193. 14$. 6d., and the remainder between 732 74 A MODERN ARITHMETIC privates. Find to the nearest penny what each private received, and the value of the remainder, if any. 42. A man buys 497 sacks of potatoes for ^339. is. n^d. and sells 248 sacks at 17 s. 9^. each, and the remainder he sells at i8s. 7.\d. each. Find his total gain. 43. If a person's income for the year 1880 be ,37. 15.9. per calendar month, and he spends at the rate of 1. 4^. 7\d. per day, how much will he have left at the end of the year ? 44. A person bought 374 eggs at 2 a penny, and some others at 3 a penny. He paid for them in all ^i. 9^. lid. How many did he buy at 3 a penny ? 13. THE MEASUREMENT OF LENGTH. In early times lengths were measured as multiples, or as portions, of different parts of the human body, this being indicated by the names which still survive, as, for example : The foot; the hand, a term used even now in expressing the heights of horses ; the span ; the pace and others. Such measurements would be rough, and vary with the person measuring; now, when by modern instruments it is easy to copy accurately a standard length, more precise methods are used. In our country, lengths are expressed in terms of multiples, or portions, of the distance between 2 marks on a rod kept at the Standards Office and copies at other places, and called the yard ; the multiples and portions in general use are the inch, the foot and the mile, connected as in the table : BRITISH MEASURES OF LENGTH. 12 inches (in.) make i foot. (A number of inches is 3 feet (ft.) i yard. frequently indicated by writ- ing down the number with 1760 yards (yds.) i mile. two accents above it thus THE MEASUREMENT OF LENGTH 75 3 inches by 3" and a number of feet by the number with one accent above it thus 2 feet by 2'; so that 2'. 4" represent two feet and four inches.) The system in use in most countries, and in much scientific work in England, is the METRIC SYSTEM, in which the unit is the Metre. The chief portions and multiples of the metre are as indicated in the table below : 10 millimetres make i centimetre. 10 centimetres i decimetre. 10 decimetres i metre. 1000 metres i kilometre. One metre is nearly 40 inches. 14. REDUCTION. In many instances it may be found necessary to change from yards to feet and inches ; from miles to smaller lengths, and vice versa. The following examples are typical of the methods employed ; EXAMPLE. Express 4 yds. 2 ft. 8 in. as a number of inches. 4 yds. 2 ft. 8 in. M Ex P ress the 4 yds. in feet, by multi- plying by 3 (as each yard is the same as 3 ft.) ; adding on the 2 ft, it is seen that 14 ft. 8 in. the 4 yds. 2 ft. is the same as 14 ft. 12 (I} Change from feet to inches in a I/7 6 j n similar manner, i.e. by multiplying the ' number which expresses the length in feet by 12, and adding on the number of inches (here 8). Thus 4 yds. 2 ft. 8 in. = 176 in. 76 A MODERN ARITHMETIC EXAMPLE. Express 5 miles 77 yards i foot as a number of feet. 5 mi. 77 yds. i foot. (a) Multiply the 5 by 1760 and so 1 760 express the 5 miles by the correspond- 8877 yds. i foot. ing number of yards -add on the "2 ' 6 (b} Reduce the yards to feet by 26632 ft. multiplying by 3, add on the i. Thus 5 miles 77 yards i foot = 26632 feet. EXAMPLE. Express 341 inches in terms of yards, feet and inches. (a) Divide the 341 by 12 to find how many feet there are in 341 12 28 ft. 5 in. inches. The result shows that there Q yds i ft c; in are 2 - anc 5 nc es over. (b) Divide the 28 by 3 to find how many yards there are in 28 ft. EXERCISES XXIII. a. Mental or Oral 1. Express in feet and inches (a) 1 6 inches, (b) 26 inches, (<r) 36 inches, (d) 50 inches, (e) 100 inches, (/) 200 inches. 2. How many inches are there in (a) 2 ft. 5 in., (b) i yd. 2 ft. 3 in., (c) 10 yds., (d) 8 yds. 2 ft, (e) 100 yards, (/) 10 yds. 4 in.? 3. How many feet are there in (a) i mile, (b) 220 yds., (c) a quarter of a mile, (d~) a mile and a half. (e) one-eighth of a mile ? THE MEASUREMENT OF LENGTH 77 EXERCISES XXIII. b. Practical. [Apparatus required: Foot-rule. Mathematical Instruments. Squared Paper.] 1. Find, in inches, as accurately as your scale permits, the length, breadth and diagonal of a page of your (i) Arithmetic text-book, (2) Arithmetic exercise book. 2. Measure the width of your desk and (either individually or collectively) the length of the desk. Make a drawing of the desk, where each foot on the desk is represented by a line, one inch, half-inch or a quarter-inch in length, as you think most suitable. 3. Set off on squared paper a line AB in one direction and a line AC at right angles to it ; make AB and AC of the lengths given below ; measure the length of BC in each case. (a) AB 3 inches, AC 4 inches, (b) AB 2 inches and 3 twelfths, AC 3 inches, (c) AB 5 eighths of an inch, AC i inch and 2 twelfths, (d) AB 3 inches and a half, AC 4 inches and 8 twelfths. EXERCISES XXIII. c. 1. How many inches are there in i mile ? 2. A train goes 60 miles an hour ; how many feet will it travel in i minute? 3. A train goes 44 feet in one second ; how many miles will it travel in one hour ? 4. What is the length in yards, etc., of 32 steam pipes, each one being 19 ft. 9 in. long? 5. The circumference of a bicycle wheel is 13 ft. 8 in. Express in miles, yards and feet the distance covered when 2764 revolutions are recorded. 78 A MODERN ARITHMETIC 6. The drum of a winding engine is 22 ft. 6 in. in circumference. What length of rope in yards is let out by 200 revolutions ? 7. A chain consists of 1742 links, each 2 inches long. What is the total length of the chain ? 8. What is the length covered by 2000 steel rails, each 23 ft. 6 in. long, placed end to end, with a total space of 20 ft. between their adjacent ends, as in a railway ? 9. Express in yards and feet the excess of I knot (6080 ft.) over a mile. 10. The highest mountain in the world is stated to be 29,002 ft. high. Express this in miles, yds. and ft. 11. A boy has used 6 balls of cord, each 145 yds. long, in flying his kite. What is the height of his kite in ft. and in. if the height is one-tenth the length of the string ? 12. The driving wheel of a locomotive engine is 23 ft. 9 in. in circumference, while the small front one is 9 ft. 6 in., how many more revolutions will the small one perform than the larger one in going a distance of 76 miles 380 yds.? 13. From the ground floor of a house to the attic are 54 steps each five inches in height: What is the height of the attic above the ground floor in feet and inches ? 14. The top of a tower is reached by 372 steps each having a rise of 4 in. What is the height of the top of the tower in yds., ft. and in.? 15. What distance does a soldier cover in a march of 4 hours, making an average of 1 10 steps each 2 ft. 3 in. per minute ? THE MEASUREMENT OF LENGTH 79 15. ADDITION, SUBTRACTION, MULTIPLICATION AND DIVISION OF LENGTHS. Addition, subtraction, multiplication and division of lengths are done by following the plan employed in similar operations in money questions, thus : EXAMPLE. Three rods have lengths respectively: 3 yards I foot 6 inches, 2 yards 2 feet 8 inches and 8 yards 2 feet 9 inches ; what is their combined length ? Arrange with yards under yards, feet under feet, etc. yds. ft. in. (a) Adding the numbers in the inches 3.1. 6 column, we get 23 in., representing i ft. 1 1 in. 2.2. 8 We set down the n in the inches column 8.2. 9 and carry on the i to the feet column. (b] Adding the numbers in the feet column, *5 to the i carried, we obtain 6, representing 6 ft. 21 or 2 yards o ft. Set the o in the feet column, carry on the 2 to the yards column. Adding this, we obtain 15. The combined length is 15 yards o ft. n inches. EXAMPLE. The length of a piece of string is 27 yards i foot 8 inches ; a piece 1 2 yards 2 ft. 9 inches is cut off ; what length remains ? As in complementary addition : yds. ft. in. (a) ii inches added on to 9 inches give 27 . i . 8 i ft. 8 in. We set down n, carrying i on to 12.2. 9 the 2 in the feet column obtaining 3 ft. ~ ~ ~ (&) i ft. added on to 3 ft. makes i yd. i ft. Carry i on to the 12 in the yards column. (c) 14 added to 13 gives 27. The length of string which remains is therefore 14 yds. i ft. n in. 8o A MODERN ARITHMETIC EXAMPLE. What is the total length of 15 balls of twine each 223 yards 2 feet 3 inches long? miles, yards, feet, inches." (a) 15 times 3 inches are 45 inches o . 223 . 2 . 3 or 3 ft. 9 inches. Set down the 9 inches, f 5 carry on the 3 ft. ~ ^77^ ~ " (If) 15 times 2 ft. are 30 ft. ; with the _ 3 ft. brought forward this becomes * 3 33 ft. or 1 1 yds. o ft. Set down the o, carry on the n. (c) 15 times 223 yards are 3345; carry on the u, and obtain 3356 yds. or i mile 1596 yds. The total length is i mile 1596 yds. o ft. 9 in. EXAMPLE. A cord 2 miles 30 yards long is divided into 24 equal parts ; what is the length of each parti 24 o miles. 147 yards. 2 ft. 9 m.Ans. ^ 2 divided by 24 gives 2 miles. 30 yards, o ft. o in. o and 2 over. I7 6o () Reducing the 2 miles to yards and adding the 30, 3550 yards, o ft. o in. we obtain (c) Dividing 3 5 50 yds. by 24, we obtain 147 yds. and 22 yards, o ft. o in. 22 yds. over. 3 (d) Reducing this 22 66 ft. o in. y ar A S l f ?* we btain 66. (e) Dividing by 24, we 1 8 ft. o in. obtain 2 ft. and 18 ft. over 12 (f) Reducing to inches, 2I 2 1 6 in. is obtained ; dividing by 24, the result is 9 in. Each piece is therefore 147 yards 2 feet 9 inches long. EXAMPLE. A board is 8 yds. 2 ft. 7 in. in length; how many pieces i yd. o //. 3 in. long can be cut off it, and what length of board will be left over ? Reducing both lengths to inches, we have to find out how many lengths of 39 inches can be cut from a board 26 ft. 7 in or 319 inches long. THE MEASUREMENT OF LENGTH 81 Dividing 319 by 39, we obtain 8 and 7 over. The board may therefore be cut into 8 pieces each i yd. o ft. 3 in. long, and a piece 7 in. long will remain. EXERCISES XXIV. 1. The three sides of a triangle are respectively 2 yds. 2 ft. 2 in,, 4 yds. o ft. ii in. and 3 yds. i ft. 10 in. What is the length of the perimeter or distance right round the triangle ? 2. In order to get from a place A to a place B, one man follows a straight road AC i mile 725 yards long, turns at C and follows another road I mile 1200 yards long. A second man takes a direct cut from A to B, and thereby saves a distance of 800 yards. What is the direct distance between A and B ? 3. How many tiles each 6 in. long will be necessary to enclose a square plot of grass, each side being 23 ft. 9 in.? 4. A wire fence 212 yds. i ft. long is fitted with uprights 2 ft. 2 in. apart. What number of uprights will be required ? 5. Each turn of a coil of wire is i ft. 7 in. long. What length of wire will be required to make 200 turns ? 6. In paving one side of a street 872 kerb stones each of an average length of 2 ft. 4 in. are used. What is the length of the street ? 7. From a roll of calico 60 yds. long, lengths of 34 yds. 8 in. and 49 ft. 9 in. are cut off. What length remains ? 8. A path in a park is edged on both sides with tiles 5 in. long. What is the length of the path when 3642 tiles are required ? 9. A short piece of railway measuring 3 miles 1482 yds. is to be made by 63 men. What length must each make ? 10. Three roads in a parish are severally 4 miles 753 yds. 2 ft., 2 miles 1628 yds., and 6 miles 905 yds. in length, and are kept in repair by 5 men. What is each man's share ? 11. Goods have to be taken a distance of 50 miles. Of this 9 mi. 1342 yds. is by canal, and 36 mi. 570 yds. by rail, and the remainder by cart. What distance is performed by cart ? J.M.A. F 82 A MODERN ARITHMETIC 12. Fill in the tables below, giving the thicknesses and depths of the strata met with in making the wells specified : (a) Well at Marylebone Road : Strata. ^Tn!' De P th ' Made ground - 3 o (b) Well at Greenwich pital : c f . Thickness. Strata. ft j n Hos- Depth. Gravel and clay 8 o Made ground - 1 1 o Gravel - - 33 o Black sand - 4 10 Blue clay - -08 Peaty earth - 02 Shelly rock - 40 Clayev sand - I 8 Red clay - -60 White sand -40 Grey ,, - 21 o Green sand and Chalk - - 178 o pebbles - -40 Thanet sand - 55 10 Flint bed - - i o (c) Well at Waltham Abbey : Chalk with flints 180 6 Strata. Th kn f n s ; Depth. Soil - - - i o (d) Well at Lea Bridge , Clay - - - 2 6 Strata. Thickness. Depth. Peat - - - 4 6 ... ft. in. Made ground - 60 River gravel - 10 o Red ballast - 12 o Blue clay - - 26 o Peat - - -20 Sandy clay - 10 6 Ballast - -70 Sand and shell - 04 Grey sand - -80 Fine hard sand - i o Loam - -06 Pebbles - -02 Sand - i o Blue clay - - 80 Dark sand - i 8 Sandy clay - 4 o White sand - 411 Dark sand and sandstone - 40 Sandy clay I i . . Conglomerate - 20 Green sand - 8 o Coloured sand - 3 o Light ,,-50 Coarse ,,-46 Fine - 27 9 Flints - o 9 Chalk - - 75 o Dark sand - 20 Grey - 21 o Very hard sand - 6 1 1 Soft green ,,-07 Hard ,,-21 Soft sand - - 2 6 Flint - - - o 10 Bog White sand >> >> gravel 6 9 10 12 12 )) Marl White sand and flints - - 30 o Grey chalk - 36 o White sand - 37 o Grey chalk - 60 i Hard grey chalk 79 10 Chalk with flints 152 10 Chalk - 202 7 Sandy marl - 203 5 Rotten chalk - 210 5 Chalk - - 250 5 THE MEASUREMENT OF LENGTH (e) Well at Dartford : (/) Well at Ealing : Yellow clay Clay with clay- stones - Clay and green sand Mottled clay - Loam Mottled clay - Hard sand Red clay - Green and black sand - Chalk and flints Chalk - Chalk and flints Hard chalk - Chalk - Grey chalk with flints - Chalk with flints Chalk Chalk with flints Grey chalk with flints 35 o - 253 6 264 o 304 o 306 o 307 3 3io 3 321 2 328 9 330 3 347 3 418 5 430 ii 480 10 488 10 500 4 501 4 538 4 - 601 8 EXERCISES XXV. Practical. [Apparatus: Scale graduated in fractions of an inch. 1. Employ your scale to find ; (a) The number of eighths in 1 5 twelfths ; (b) The number of eighths in 27 twelfths ; (c) The number of twelfths in 14 eighths ; (d) The number of quarters in 33 twelfths. Paper.} S 4 A MODERN ARITHMETIC 2. Draw any four parallel straight lines (or take any four which are ruled on the paper) at equal distances apart. Draw a straight line meeting the four parallel lines, and letter the points of meeting A, B, C, D taken in order ; measure AB, BC, CD. Repeat with five other lines like ABCD ; measure the parts cut oft between the parallel lines in each case, and state what you notice about these parts. 3. Draw a line 4 inches and 7 twelfths, divide into five equal parts, measure the length of one of these parts. 4. Use your scales to find the sum of 2 inches, 3 inches and a quarter, I inch and 5 eighths, 2 inches and a twelfth. 5. Copy accurately the accompanying figures, and mark on them the dimensions of the various lines. FIG. 10 (a). FIG. 1 . 6. The accompanying figure represents the two faces of an ordinary brick, which meet along one edge. FIG. ii. (The figure should, however, be enlarged six times to give a true representation). Draw a figure representing the other face which meets the two given faces at the corner A, and write down the length, breadth and thickness of the brick. REDUCTION IN METRIC OR DECIMAL SYSTEM 85 16. REDUCTION IN THE METRIC OR DECIMAL SYSTEM. In this system the process of reduction is very simple, owing to the multiples and submultiples all being powers of 10. Thus : EXAMPLE. To express 3 metres 4 decimetres 2 centimetres 8 millimetres in millimetres. It is usual to abbreviate by representing metres by m., decimetres by dm., centimetres by cm. and millimetres by mm. .'. 3 m. 4 dm. 2 cm. 8 mm. have to be expressed in mm. 3 m. 4 dm. 2 cm. 8 mm. = 34 dm. 2 cm. 8 mm. Since 3 m. are 30 dm. = 342 cm. 8 mm. 34 dm. 340 cm. = 3428 mm. 342 cm. 3420 mm. EXAMPLE. To express 7812 mm. in metres, decimetres, etc. 7812 mm. = 781 cm. 2 mm. = 78 dm. i cm. 2 mm. = 7 m. 8 dm. i cm. 2 mm. EXERCISES XXVI. a. Practical [Apparatus: Dividers. Scale on English and Metric systems. String^ 1. Measure the length and breadth of your book with your scale, writing the measurements down : (a) In inches ; (b) In decimetres, centimetres and millimetres ; (c) In centimetres and millimetres ; (d) In millimetres. 2. In the same way obtain : (a) The width of the desk ; (b) The length of your pen ; (c) The length and breadth of your exercise book. 86 A MODERN ARITHMETIC 3. Set off on your exercise books lines of the following lengths : (a) 8 cm. 3 mm., (/') 57 mm., (c) i dm. 2 cm. 5 mm., (d) i dm. o cm. 5 mm., (e) 2 dm. 3 cm. 8 mm. 4. Use your scale to find : (a) The difference between i dm. and 6 cm. 5 mm. (b) The sum of 8 cm. 4 mm. and 4 cm. 8 mm. 5. Cut pieces of string or tape of the following lengths : (a) i metre, (b) i metre 2 dm., (c) 2 metres 3 dm. 4 cm. 6. Find the number of inches in your string of i metre length. 7. Cut off a piece of string or tape of length equal to the difference between a metre and a yard ; express this difference : (a) In inches ; (b) In cm. and mm. ; (c) In millimetres. 8. Find from your scale the number of dm., cm. and mm. in one foot. EXERCISES XXVI. b. Mental or Oral. (A) Express in millimetres : 1. 6 cm. 2. 6 dm. 3. 6 m. 4. 6 m. 6 cm. 5. 6 m. 6 dm. 6. 6 m. 6 dm. 6 cm. 7. 7 rn- 3 dm. 8. 7 m. 3 mm. 9. 3 dm. 3 mm. 10. 5 cm. 2 mm. 11. 18 m. 12. i m. i cm. 13. i m. i dm. 2 cm. 14. 6 m. 2 cm. 15. 6 m. 2 mm. 16. 4 m. 2 dm. 17. 4 m. 3 dm. 2 cm. 18. 6 m. 6 mm. 19. 7 m. 8 cm. 20. 8 m. 2 cm. 21. 8 m. 3 cm. 2 mm. 22. 9 m. 9 cm. i mm. 23. 2 m. 2 cm. 9 mm. 24. 2 dm. 3 cm. 8 mm. 25. i6m. 8dm. 3mm. 26. 5 m. 3 dm. 2 mm. 27. 9 m. 2 cm. 7 mm. (B) Express in metres, decimetres, etc. : 1. 76 mm. 2. 760 mm. 3. 706 mm. 4. 8260 mm. 5. 98 cm. 6. 5309 mm. 7. 234 dm. 8. 9102 mm. 9. 912 mm. 10. 9oi2mm. 11. 8716 mm. 12. 876cm. 13. 87 dm. 14. 962 mm. 15. 8345 cm. 16. 8345 mm. 17. 8046 mm. 18. 7623mm. 19. 926cm. 20. 1763 mm. REDUCTION IN METRIC OR DECIMAL SYSTEM 87 (c) State in metres, decimetres, etc., the lengths indicated by the figures marked with an asterisk (*) 1. 34 2 ^ mm. 5. 72 1 cm. 2. 3428 mm. 6. 6103 cm. 3. 3428 mm. 7. 7 1 23 cm. 4. 10862 mm. 8. 835 mm. 9. 7273 mm. 13. 756dm. 17. 3137 cm. 10. 8342 dm. 14. 3602 mm. 18. 8106 dm. 11. 8312 mm. * 15. 7432 cm. * 19. 3134 mm. 12. 8301 mm. 16. 8155 mm. 20. 81752 dm. 21. 81752 cm. 22. 2321 mm. In the metric system, it will be noticed that multiplying or dividing lengths by 10 is equivalent to changing the denomina- tions in terms of which the lengths are expressed. EXAMPLE (i) 3 cm. x 10 = 3 dm. (2) 3 cm. -- 10 = 3 mm. Again, m. dm. cm. mm. m. dm. cm. mm. O. 3. 4. 2 X 10 = 3. 4. 2.0 5.2.5.0 -rIO = 0.5.2.5 EXERCISES XXVII. Write down the following lengths, perform the multiplications indicated and arrange as above : m. dm. cm. mm. m. dm. cm. mm. 1. O . O . O . 2 X 10 = 2. O . O . O , , 2 X 100 = 3. . . O . 2 X IOOO = 4. O . O . I , , 2 X 100 = 5. . 2 . 3 - O X 10 = 6. O . 2 . O , - 5 X 100 = 7. . . 4 4 X 1000 = 8. O . 4 4 X 100 = 9. O . 4 . o . 4 X 10 = 10. 2 . 3 2 . X 10 = 11. O . o . i . 7 X 100 = 12. 3 4 5 - X 10 = 13. 13 . i . 3 o X 10 = ss A MODERN ARITHMETIC Divide the following : m. dm cm. mm. m. dm. cm. mm. 14. I . O . O O - - 10 = 15. I . O . O . -f- 100 = 16. I . O . O o - -1000 = 17. 3 . 2 . O . O -7- 100 = 18. 2 3 3 o - - 10 = 19. 2 3 . O . -T- 10 = 20. 2 . 3 o - IO = 21. 23 . i . . -f- 100 = 22. 106 . I . 2 o - IO = 23. 1 06 . 7 . . -T- 100 = 24. 24 . I 3 o - - 10 = 25. 57 4 . . -f- IO = 26. 88 . 8 . 4 o - - 10 = 27. 7 . . 6 . -r- IO = 17. INTRODUCTION OF DECIMAL NOTATION. We have seen that a length like 2 m. 3 dm. 4 cm. 2 mm., can be written also 234 cm. 2 mm. Here, considering the figures representing the number of centimetres, the 3 in the tens place represents 3 decimetres or 30 centimetres, i.e. its value is 10 times as great as it would be if it had been located in the units place. Similarly, the 2 in the hundreds place has a value 100 times as much as if it were written in the units place, and ten times as much as if it had been written in the tens column. The length represented by any of the figures depends upon the place of the particular figure. Further, the 2 in the mm. column represents a length one-tenth of what it would be were the 2 placed in the units place in the centimetre column; and the length 234 cm. 2 mm. may be expressed in terms of cm. by adopting some device to indicate the unit figure. This is done by placing a decimal point immediately after the units figure so that 234-2 cm. means 2 hundreds of centimetres 3 tens 4 units INTRODUCTION OF DECIMAL NOTATION 89 and the 2 following the decimal point has one-tenth the value it would have if placed immediately to the left of the decimal point, i.e. 2 millimetres or 2 tenths of a centimetre. The notation is extended still further to the right of the decimal point, and thus : 34-24 dm. is the same as 3 metres 4 dm. 2 cm. 4 mm. Although the examples given are all connected with lengths expressed in the metric system, the decimal notation can be applied to all kinds of quantities. Thus, 34-24 ft. means 34 feet 2 tenths of a foot and 4 hundredths of a foot. Again in 36-63 lb., the 6 after the decimal point represents just one-tenth the mass that the 6 before the decimal point represents, while the first 3 refers to a mass one-thousand times the 3 appearing in the 2nd decimal place. Moreover, as in the case of ordinary numbers, we can make use of decimals in the abstract without reference to any concrete quantities at all. Thus 0-3, 0-5, 0-6 ; and 0-3 is one tenth of 3, 5 is ten times 0-5, 6 is ten times 0-6. EXERCISES XXVIII. {These exercises may be worked through orally first, and then again considered as 'written work.) Express : 1. 3 m. 2 dm. 8 cm. as metres and decimals of a metre. 2. 0-416 metre as (a) decimetres, () centimetres and (c) milli- metres. 3. 7 m. 2 cm. 5 mm. as (a) decimetres, (b) centimetres and (c) millimetres. 90 A MODERN ARITHMETIC 4. 14 m. 6 cm. 9 mm. as metres and decimals of a metre. 5. 29 cm. 6 mm. as metres and decimals of a metre. 6. 1 32 dm. 3 cm. 6 mm. as metres and decimals of a metre. 7. 84750 metres as (a) decimetres, (b) centimetres and (c) milli- metres. 8. 0-718 m. as cm. 9. 0-071 m. as cm. 10. 0-007 m - as cm - 11- 3' 1 7 dm. as cm. 12. 3 m. 2 cm. as metres. 13. 6 m. 2 dm. as metres. 14. 3 m. 2 dm. 2 mm. as metres. 15. 4 m. 3 mm. as metres. 16. 3 m. 7 dm. 3 mm. as dm. 17. 2 m. 2 mm. as dm. 18. 3 m. 4 mm. as cm. 19. 2 m. 3 dm. 2 mm. as cm. 20. 0-36 dm. as cm. 21. 0-412 m. as mm. 22. 0-018 m. as cm. 23. 4-16 m. as mm. 24. 6 m. 2 cm. as mm. 25. 6 m. 2 cm. as dm. 26. 6124 cm. as m. 27. 0-012 m. as cm. 28. 4-12 dm. as mm. 29. 3 m. 2 cm. 4 mm. as metres. 30. 1-260 m. as mm. Express, as asked for, the lengths indicated by the figures marked with an asterisk (*) in the following : * * 31. 34-24 cm. in cm. 32. 34-24 cm. in mm. * * 33. 53-24 m. in cm. 34. 53-24 in. in mm. * * 35. 53-24 m. in cm. 36. 6-25 metres in mm. * * 37. 17-32 metres in mm. 38. 80-01 cm. in mm. * * 39. 800- 1 cm. in metres. 40. 6-23 dm. in cm. How many times greater is the quantity represented by the figure marked a than the length represented by the figure marked ? a b a. b 41. 3-424 metres. 42. 24-24 metres. a b a b 43. 24-24 miles. 44. 60-06 miles. INTRODUCTION OF DECIMAL NOTATION 91 45. 60-06 . 46. 0-863 ;. 47. 36-63 . 48. 1-234 yards. a. b a b 49. 4-321 pounds. 50. 1-065 gallons. 18. ADDITION AND SUBTRACTION OF METRIC LENGTHS. EXAMPLE. One string is 4 metres 2 decimetres 3 centimetres 8 millimetres in length, another string is 6 metres 8 decimetres 2 centimetres 4 millimetres long, and a third is 8 metres 7 decimetres 7 centimetres 8 millimetres ; what is the combined length ? The example may be set down as in the addition of length in yards, feet and inches, thus : m. dm. cm. mm. 4.2.3.8 (8 + 44-8) mm. are 20 mm., i.e. 2 cm. o dm. 6.8. 2.4 (2 + 7 + 2 + 3) cm. are i4cm.,z'.. i dm. 4 cm. 8.7.7.8 (i + 7 + 8 + 2) dm. are 1 8 dm., i.e. i m. 8dm. (1+8 + 6 + 4) m - are 19 m. 19 . 8 . 4 . o The combined length is therefore 19 m. 8 dm. 4 cm. o mm. The example may be worked more conveniently in terms of one denomination and decimals, thus : EXAMPLE. (i) Expressing in metres. 4-238 6-824 8-778 (2) Expressing in dm. 42-38 68-24 87-78 (3) Expressing in cm. 682-4 877-8 1984-0 cm. (4) Expressing in mm. 4238 6824 19-840 metres. 198-40 dm. 19840 mm. Care must be taken that the decimal points are all under one another, in order that units are under units, tens under tens, etc. 92 A MODERN ARITHMETIC EXAMPLE. A rod is 3 m. o dm. 2 cm. 8 mm. long; it is cut into two pieces^ one of which is i m. 2 dm. o cm. 9 mm. long. What is the length of the other piece ? Here it is required to subtract i m. 2 dm. o cm. 9 mm. from 3 m. o dm. 2 cm. 8 mm. m. dm. cm. mm. 3.0.2.8 1.2.0.9 I.8.I.9 The remaining piece is therefore i m. 8 dm. i cm. 9 mm. Or, working in decimals of a metre we subtract 1-209 from 3-028, setting decimal points underneath one another for the same reason as in addition. 3-028 1-209 The remaining length is seen to be 1-819 metres. 1-819 EXEECISES XXIX. . j m. dm. cm. mm. 1- Add 3.2.0.6 0.0.8.2 1.0.7.6 2.3.0.1 Set down the lengths as decimals of a metre also, and add up again, checking the two results by one another. Find the following sums, verifying as in the Example above. 2. 3 m. 3 dm. 3 cm. 3 mm., 4 m. 2 dm. 2 cm. 6 mm., 4 m. 2 dm., 4 m. 2 cm. 3. 15 m. 2 cm. 3 mm., 2 cm. 3 mm., 6 m. 3 dm. 8 cm., 8 m. 2 cm. 4. 6 m. 6 cm. 6 mm., 2 mm., 8 m. 8 cm. 8 mm., 9 m. 4 cm. 5 mm., 6 cm. 4 mm. 5. 23 m. 4 dm. 5 cm. 6 mm., 4 dm. 5 mm., 9 cm. 8 mm., 7 m. 4 cm. 3 mm., 7 m. 2 dm. 5 cm. 4 mm. 6. 8 m. 3 cm. 3 mm., 3 m. 8 cm. 8 mm., 6 m. 5 cm., 8 m. 9 dm. 7 cm. 7 mm. ADDITION AND SUBTRACTION OF METRIC LENGTHS 93 7. ioo m. 2 cm. 6 mm., 8 m. 9 dm. 7 cm., 24 m. 8 dm., 3 m. 2 dm. 3 cm. 8. 7 m. 6 dm. 2 mm., 18 m. 9 dm. 6 cm. 5 mm., 13 m. 4 cm. 5 mm., 12 m. 2 dm. 9 min. 9. ii m. 8 mm., i m. i dm. 8 cm., 11 m. 8 cm., 11 m. 8 dm., 1 dm. i cm. 8 mm. 10. 9 m. 5 dm. 3 mm., 3 m. 5 cm. 9 mm., 13 m. 8 dm. 7 mm., 13 m. 7 cm., 7 dm. 8 mm. 11. 26 m. 2 mm., 26 m. 2 cm., 26 m. 2 dm., 2 m. 6 dm. 2 mm., 2 dm. 6 cm. 2 mm. 12. 19 m. 8 dm. 6 mm., 19 m. 2 mm., 37 m. 8 dm., 3 m. 8 dm. 9 cm., 3 m. 8 dm. 9 mm. 13. 49 m. 5 mm., 6 dm. 7 mm., 8 m. 7 dm., 15 m. 8 dm. 7 cm. 2 mm., 15 m. 8 cm. 7 mm. 14. 5 m. 8 dm. 6 cm., 6 m. 8 dm. 5 mm., 6 dm. 8 cm. 5 mm., 6 m. 8 dm. 5 mm. In Exercises 15-34 perform the subtraction operations indicated, verifying by working also in decimals of a metre : m. dm. cm. mm. m. dm. cm. mm. 15. From 12 . 2 3 4 take 4 . 3 . 2 . 6 16. 8 . 8 . o . 8 55 8 . o . 8 . 8 17. 5) 9 . . . o 7 7 . 6 3 18. 5) 26 . I . 8 . 3 55 21 . 8 . i 3 19. 55 103 4 . 6 . 7 55 26 . 8 3 . 8 20. 55 IOO . 2 . i . 2 55 3 - 8 9 9 21. Subtract 3 m . 2 dm. 2 mm. from 6 m. 2 mm. 22. 55 4 m. 2 dm. 3 cm. 5 mm. 55 10 m. 23. 55 18 m. 3 cm. 3 mm 55 38 m. I dm. i mm. 24. 55 7 m. 2 cm. i mm 55 10 m. 8 dm. 25. 51 6 m. 6 dm. 6 cm. 6 mm. 55 8 m. 2 dm. 3 mm. 26. 5J 15 m. 6 dm. 3 cm. 2 mm. 55 18 m. 3 cm. 3 mm. 27. 55 19 m. 3 cm. 55 19 m. 3 dm. 28. 15 7 m. 4 dm. 2 cm. 2 mm. 55 12 m. 2 mm. 94 A MODERN ARITHMETIC 29. Subtract 4 dm. 3 cm. 2 mm. from 1 1 m. 30. 3 dm. 8 mm. 4 m. 8 cm. 31. 2 m. 7 cm. 6 mm. 3 m. 2 cm. 32. 5 m. 6 dm. i mm. 6 m. 5 dm. I cm. 33. ,, 7 m. 6 dm. 7 cm. 7 mm. 12 m. 8 dm. 8 cm. 8 mm. 34. 23 m. 7 dm. i cm. 2 mm. 30 m. 3 dm. I mm. 35. Find the sum of 0-896, 2-175, 3 2>O1 and 6-08. 36. The weights of coal in four consecutive trucks in a train are respectively 8-27 tons, 10-08 tons, 12-31 tons, 7-89 tons ; what total weight is contained by the four trucks ? 37. Five rods are placed in line with one another, their respective lengths being 14-26 ft., 13-89 ft., 15-72 ft, 13-72 ft. and 16-03 ft. ; what is their combined length ? 38. A piece 3-26 ft. is sawn off a rod 12-37 ft. long ; what is the length of the remainder ? 39. A vessel contains 36 gallons of water, 7-83 gallons of which are run out ; what does the vessel now contain ? 40. Evaluate : (i) 6-32 + 7-i6 + o-87 + o-26 + 7-8i4. (ii) 216-1 +3-814 + 0-072 + 283-1 +3-141 +0-082. (iii) 79-3 + 9I-6 + 90-07 + 0-92+ 167-3 + 0-89. (iv) 7-36+ 18-02 + io6-35+o-8i6 + 29-3 + 3o-oo5. (v) 236-2 + 0-89+ 107-341 +8-36 + 0-97+ 13-776. (vi) 378-342-186-593. (vii) 612-7-89-821. (viii) 106-82-101-61. (ix) 81-932-11-75. (x) 119-171-71-91. (xi) 379-23 -193-45. (xii) 100-0-09. (xiii) 16-342+17-95- 18-082-0-93. (xiv) 102-61 +92-83 -10-98 -83-62. (xv) 37-86-79-21+64-18. (xvi) 116-375-182-02 + 217-362. (xvii) 89-106 + 3-782 -0-983 + 2-17. (xviii) 108-94 -893-06 + 0-89 + 972-5. (xix) 383-16 + 82-76-500+142-803. MULTIPLICATION AND DIVISION OF DECIMALS 95 19. MULTIPLICATION AND DIVISION OF DECIMALS BY A SINGLE FIGURE. EXAMPLE. Six rods each 4 metres 3 decimetres o centimetres 2 millimetres in length are placed end to end. What is the combined length ? m. dm. cm. mm. 4.3.0.2 6 25 . 8 . i . 2 The combined length is therefore 25 metres 8 decimetres i centimetre 2 millimetres. Working in decimetres and decimals of a decimetre, we proceed thus : dm. 43-02 . 6 258-12 The combined length is 258-12 decimetres. EXAMPLE. What is the combined weight of 7 b locks , each one of which weighs 14-038 Ibs. ? By addition. By multiplication. 14-038 I4-038 14-038 7 14-038 08-266 The procedure is as in ordinary 14-038 multiplication ; only, as we pass 14-038 the decimal point, it is set down 14-038 in the answer. 14-038 98-266 The total weight is 98-266 Ibs. 96 A MODERN ARITHMETIC EXAMPLE. A rope 192 metres 7 decimetres 8 centimetres 6 millimetres long is divided into six equal parts. What is the length of each part ? m. dm. cm. mm. 6 192 .7.8.6 ?tres + 1,0 32 Or, working 'with lengths expressed in metres and decimals of a metre, m. 61 192-786 32-131 The length of each of the portions is 32-131 metres. EXAMPLE. If 272-365 gallons be divided into five equal parts, what will be the volume of one of these parts ? 54-473 Each part therefore has a volume of 54-473 gallons. EXERCISES XXX. 1. Multiply 8 metres 2 decimetres 3 centimetres 4 millimetres by 8 ; also express the length in metres and decimals of a metre and multiply the resulting number by 8. 2. Multiply as in (i) : m. dm. cm. mm. m. dm. cm. mm. (i) 7 4 3 2 by 7- (ii) 12 3 2 5 by 9- (iii) 8 . 2 3 6 ii. (iv) H 3 . 9 . 7 ,, 6. (v) o . 3 7 6 4- (vi) 8 . 6 . 8 . i ii 3- (vii) 13 7 5 2 7- (viii) 8 . 6 . 2 . 5 i, 8. (ix) 9 i . 2 . 5 8. (x) 1 1 7 7 5 11 4- (xi) 8 . 3 7 5 4- (xii) 10 . i . 2 . 8 12. MULTIPLICATION AND DIVISION OF DECIMALS 97 3. Divide 37 m. 2 dm. 3 cm. 6 mm. by 6 ; also express the length as a decimal of a metre, and divide this number by 6. 4. Divide as in (3) : m. dm. cm. mm. m. dm. cm. mm. (i) 2 . 6 . 3 . 5 by 5. (ii) 6 . 3 . 8 . o by 4. (iii) 3.1.4.1,, 9. (iv) 5 . 3 . 7 . 6 6. (v) 9.0.2.0,, ii. (vi) 7.1.3.1,, 3. (vii) 4.9.6.5,, 5. (viii) 8.3.2.3,, 7. (ix) 2.6.2.4,, 8. (x) 36 . 5 . i . 3 i, 9- (xi) 6.0.0.0,, 8. (xii) 21 . 2 . 3 . 4 3. (xiii) 16 . 5 . 6 . 4 4. (xiv) 2.0.7.5,, 5. (xv) 8.9.7-6,, 12. 5. In lifting a weight by a block of pulleys, the weight only rises one-fifth of the distance the rope is pulled down by the man lifting the weight. What distance will the weight be lifted when the man pulls in 6-75 ft. of rope? State also the length of rope pulled in by the man when the weight is lifted through a distance of 3-28 ft. 6. Multiply : (i) 8-961 by (a) 3, (*) 5, (c) 7, (rf) 9- (ii) 0-0625 by (a) 2, (b) 4, (c) 6, (d) 8. (iii) 14-016 by (a) 7, () 9, (c) 8. (iv) 3-162 by (a) 4, (d) 3, (c) 7, (d) 9. 7. Divide : (i) 7-6 by (a} 2, (b) 4, (c) 8. (ii) 0-0621 by (a) 3, (6) g. (iii) 13-629 by (a) 3, (b) 7- (iv) 624-375 by a (5), b (3), r (9). (v) ibr(a)2,(*)4.(f)a (vi) 3-0 by (a) 4, () 8. (vii) 411-264 by () 4, (b) 3, (*) 12. (viii) 0-0024 by (a) 3, (J) 4, (*) 6, (<*) 8, (*) 12. (ix) 0-61782 by (a) 3, () 6. (x) 36-0024 by (a) 2, () 4, fc) 8. (xi) 7-2 by (a) 5, 0)12, (08, (<09- J.M.A. G 98 A MODERN ARITHMETIC 8. What is the sixth part of (a) ,4-7268, (b) 18-324 Ibs., (c) 170-8146 ft? 9. Subtract the seventh part of ,8-743 fr m t^ e twelfth -part of ^36-3 2 4. 10. Add the eleventh of 89-32 Ibs. to the fifth part of 9-3 Ibs. 20. MULTIPLICATION OF DECIMALS. General Case. When a number, say 328-25, is multiplied by 10, the effect, as has been seen, is that where we had 3 hundreds we now have 3 thousands, 2 tens ,, 2 hundreds, 8 units 8 tens, 2 tenths 2 units, 5 hundredths ,, 5 tenths. The place value of each digit is raised tenfold ; the result is thus represented by 3282-5, and each digit is moved one place to the left with reference to the decimal point. If we multiply by 100, the units become hundreds, the tens become thousands, tenths become tens, and hundredths become units ; the place value of each digit is raised a hundredfold; the result 32825 is obtained by moving each digit two places to the left. When, in a similar way, we multiply by o-i, each unit becomes a tenth, and following the analogy each ten becomes a unit ; so that in the product of 328-25 by o-i, instead of 3 hundreds we have 3 tens, 2 tens 2 units, 8 units 8 tenths, 2 tenths 2 hundredths, 5 hundredths ,, 5 thousandths. MULTIPLICATION OF DECIMALS 99 The place value of each digit is reduced to one-tenth of its value, so that 328-25 when multiplied by o-i becomes 32-825; and we notice that all the digits are moved one place to the right relatively to the decimal point. Similarly, multiplying by o-oi, the place value will be further reduced, so that thousands digits become tens digits, hundreds digits become units, tens digits become tenths digits and so on; and 328-25x0-01 becomes 3-2825; each figure being moved two places to the right. EXAMPLE. To multiply 328-25 by 0-4. The result is evidently 4 times that of multiplying by o-i, reducing the place value of each digit one-tenth and multiplying by 4 ; in other words, multiply by 4 and move each digit one place to the right. EXAMPLE. To multiply 328-32 by 0-04. The place value of each figure is reduced to a hundredth of the former value, and the altered number then multiplied by 4. Thus, 328-32 x 0-04 becomes 3-2832 x 4= 13-1328. EXERCISES XXXI. a. Mental or Oral. 1. (a) Multiply 1000 by 4, 0-04, 0-4. (b} 300 by 30, 0-03, 0-003, '5> 0-06. (c) 250 by 0-4, 0-2, 0-02, 0-008, 12, 1-2. (d) 800 by 0-05, 0-005, 5 5 -o7 5 0-9. (e) 0-05 by 10, loo, 300, o-oi. (/) 37-2 by o-i, o-ooi, o-oi. (g) i -6 by 0-3, 0-9, 9, 0-09. 2. One line is 3 m. 2 dm. i cm. long, a second line is 0-4 times as much ; what is its length ? ioo A MODERN ARITHMETIC 3. What is the length of a line 0-07 times that of a line 83 m. 2 dm. long? 4. 0-4 of a street 378 m. 4 dm. long has to be asphalted and the remainder to be paved. What lengths have to be treated in each way ? EXAMPLE. To multiply 26-3 by 24-52. Following the reasoning employed in ordinary multiplication, we may say that we have to find (26-3 x 20) + (26-3 x 4) + (26-3 x 0-5) + (26-3 x 0-02). The arrangement is as follows : 26- (a) Multiply by 20 by first increasing the place 24-52 value by 10 (moving figures one place to left), and then multiply the result by 2, obtaining 526. 3 . (b) Multiply by 4 and obtain 105-2. 13-15 (f) Multiply by 0-5, i.e. move the figures one place 526 to right relatively to the decimal point, and multiply by 5, obtaining 13-15. 44 ' (d) Multiply by 0-02, by moving the figures two places to the right relatively to the decimal point, and multiplying by 2, obtaining 0-526. (e) Add up 526, 105-2, 13-15, 0-526, obtaining as final result 644-876. We can conveniently arrange the work as below, where it will be noticed that 2Q - (a) The units digit of the multiplier is placed 2 4-S2 un der the last figure of the multiplicand. NOTE When the multiplier is less than unity, 526- say 0-24, this cypher, which should always precede 105-2 the decimal point, must take the place of the units I 3 >1 5 digit, and be placed under the last figure of the 5 26 multiplicand. 644-8 76 () When we multiply by any digit, the partial - product is set down with its figure of lowest place value immediately under that digit, i.e. in multiplying by 4, the 2 is set underneath the 4 ; in multiplying by 0-02 in the last partial product, the 6 is underneath the 2 and so on. MULTIPLICATION OF DECIMALS 101 (<r) The decimal point in the answer is now underneath the decimal point of the multiplicand before the multiplication took place. (d) There are as many decimal figures in the product as there are altogether in the two numbers multiplied ; thus in the example we have two decimal figures and one decimal figure respectively in the two numbers whose product is found, and three, i.e. 2 + 1 decimal figures in the answer. EXERCISES XXXI. b. 1. If one cubic foot of fresh water weighs 62-5 Ibs., what will be the weight of I i-i I cubic feet ? 2. If a gallon equals 0-16 of a cubic foot, what will be the weight of a gallon of water, assuming one cubic foot of water to weigh 62-5 Ibs.? 3. If sea water be 1-027 times as heavy as fresh water, and one cubic foot of fresh water weighs 62-5 Ibs., how much will a cubic foot of sea water weigh ? 4. If iron be 772 times as heavy as fresh water, what is the weight of a cubic foot of iron ? 5. An acreage of 27-75 acres is rented at ,12-8 per acre. What is the rent ? 6. An engine draws up from a pit 376 trams of coal each averaging 0-645 tons. What is the total amount raised ? 7. A liner completes a voyage in 6-4 days, and makes an average run of 18-25 knots per hour. What distance has been run, assuming a knot to be 1-15 miles ? 8. What is the value to the nearest franc of 79-24 metres of silk at 8-75 francs per metre? 9. The wheat yield of a field of 19-6 acres is at the rate of 27-45 bushels per acre. What is its value at ^0-325 per bushel? A MODERN ARITHMETIC I 10. If 0-917 of a sovereign be pure gold, how much pure gold is there in 23-27 Ibs. of standard gold ? (Standard gold is the gold from which sovereigns are made.) 11. If the weight of a sovereign when turned out from the mint be 123-27 grains, what is the weight of pure gold that a sovereign contains ? (Refer to previous question.) 12. If ^i be equivalent to 25-25 francs, what sum in francs would be equivalent to (a) ,3-5, (b) 13-42 ? 13. If one foot of lead piping weighs 1-82 Ibs., what will be the weight of a piece of such piping 22-3 feet in length ? 14. If i Ib. of water has a volume of 0-016 cubic feet, what will be the volume of an amount of water which weighs 26-2 Ibs.? 15. When a body floats it displaces its own weight of water. What volume of water will be displaced by a floating body whose weight is 7-26 Ibs.? (i Ib. of water occupies a volume equal to 0-016 cub. ft.) 16. What will be the volume of water displaced by a boat weighing 420 Ibs.? (Refer to previous question.) 17. Assuming i metre = 39-37 inches, find the English equivalents in feet and inches of the following lengths : (a) 5 m. (V) 3 m. 2 cm. (^) 3 m. 2 dm. (d) 3 m. 2 mm. (e) 36 m. 3 mm. (/) 78-32 cm. (g) 91-82 cm. (k) 101-63 mm - 00 238cm. ( 2-707 dm. () 0-5885 m. (/) 101-2 mm. NOTE. i metre is not exactly 39-37 inches, so that the results obtained in this exercise are not quite in accordance with the truth. EXERCISES XXXI. c. 1. Multiply 763 by (a) 7-26, (b) 0*726. 2. 893-21 by (a) 0-0076, (b} 76, (c) 0-076, (d) 2-82, (*) 0-282, C/)3i7, ()3-i7. MULTIPLICATION OF DECIMALS 103 3. Multiply 0-0935 by (a) 87-64, (b) 876-4, (c) 8764, (d) 590, (e) 5900. 4. 100-82 by (a) 375, () 375. fc) -375> (<0 I3'86, (*) 1-386. 5. 97-875 b y (*) 424, () 4-24, fc) 42-4, 00 800, (<?) 80, (/) 0-008. 6. 68-125 by (a) 128, () 12-8, (<r) 0-0128, (<af) 1728, (e) 17-28, (/) 172-8. 7. 89-926 by (a) 212-5, 0*) 21-25, (0 2-125, (^) 0-02125, 8. Evaluate (a) (76-32)2, (J) (8- 4 ) 3 , (f) (o-8 5 ) 3 , (rf) (i- 7 5) 3 - 9. () (7-6) 2 x (2.1)2, ( ^ (98)2 x (o . 025)2j (c} (I . 2)3 x (o . 04)2< 10. (a) (io-2) 2 , (b) (io-3) 3 , W (100.5)2, (rf) (500-1)2. 11. (a) (896-2)2, (^) (86-21) x (0-762), (c) (98-2) x (9-1). 12. (a) (1001-2) x (998-8), (b) (100-25) x (9975), (')(&> i) x(79-9). EXERCISES XXXI. d. Mental or Oral. Write down the product of the numbers represented by the digits marked with an asterisk. * * * * 1. 432-12 and 0-062.. 2. 59-82 and 2-32. 3. 5-86 and 102. 4. 37-402 and 0-0372. 5. 374-02 and 0-372. 6. 0-7182 and 718-2. 7. 0-572 and 0-572. 8. 888-2 and 2-88. * * 9. 1003-5 an d 0-001035. 104 A MODERN ARITHMETIC EXERCISES XXXI. e. Practical. \Apparatus : Calipers, blocks, coins, tubes, spheres and scales."] 1. Employ your calipers, or other instruments, to measure : (a) The diameter of a halfpenny. (b) The diameter of a penny. (c) The dimensions of the given block. (</) The inside diameter of the given tube. Express your measurements in inches, and also in centimetres and millimetres. 2. Measure the diameter of the given sphere. To measure the diameter of a ball an indirect method must be used, because of the difficulty of measuring the diameter of a sphere directly with a scale. th FIG. 12. Measurement of the diameter of a sphere. Procure two rectangular wooden blocks larger across than the sphere. Push the two blocks against a third, or against any upright surface, so as to keep them parallel, and then place the wooden sphere between them and measure its diameter, as in Fig. 12. The scale is there shown placed on the blocks "edge on," which is the correct way to use a wooden scale for such measurements as this. A thin metal scale, however, may be laid flat on the blocks. Make measures with the sphere in different positions, and find the mean or average result. Measure the diameter of the sphere also by means of calipers. MULTIPLICATION OF DECIMALS 105 3. (a) Draw 9 parallel lines at a distance apart of one inch, and number them o, i, 2, ... 8. Take any point O on the 'o' line and draw a straight line OA to the ' 8 ' line ; measure along the * 8 ' line, a distance AB of one inch, and join OB. Mark the intercepts on your lines i, 2, etc., by OA and OB, (i), (ii), (iii), (iv), etc., respectively ; measure the intercepts, and test them by the scale of eighths. FIG. 13. (b) Draw a new set of 9 parallel lines, at any distance apart, provided the lines are equidistant, and repeat the procedure in 3 (a). [The results of the class may be compared on the board.] (c) Repeat the procedure in 3 (a), using however u straight lines, and again test the lengths of the intercepts. The gradual increase of length of the intercepts suggests a means of measuring small distances by means of a wedge. 4. Take a strip of squared paper, divided into centimetres, and cut from it a right-angled triangle with a base 10 cm. long and the perpendicular i cm. long. FIG. 14. Using a glass tube, push the acute angle of the wedge into the tube until the wedge will go no farther without getting bent, then read off the diameter of the tube. 5. Construct a similar wedge, but where the height is one inch and the length 10 inches. (a) Measure the diameter of the tube to the hundredth part of an inch. io6 A MODERN ARITHMETIC (b) Express in cm. : (i) o-i in., (ii) 0-16 in., (iii) 0-2 in., (iv) 0-3 in., (v) 0-36 in. (c) Express in inches : (i) i cm., (ii) 1-8 cm., (iii) 0-9 cm., (iv) 1-5 cm. (v) 2-1 cm. 6. Repeat, if necessary, Exercise 3 (a) and 3 (b\ and notice that, if in each case the intercepts be produced a further distance of one inch, (i) all the ends will still lie on a straight line ; (ii) the lengths will now be (a) one inch, one inch and an eighth, one inch and two eighths, etc, ; (b) i inch, i-i inch, 1-2 inch, 1.3 inch, etc. FIG. 15. Study a diagonal scale ; notice that you can read off distances as small as o-or inch and rising by steps of o-oi inch to 5 inches. 7. Use a diagonal scale to draw lines of the following lengths : 0-82 in., 0-28 in., 1-28 in., 1-82 in., 4-82 in. 4-28 in., 2-56 in., 2-65 in., 3-65 in., 3-56 in. 8. Calculate in each case the length in inches equivalent to i cm., 4 cm., 6 cm., 8 cm., 5 cm., 3 cm. 9. If a straight line measured from left to right be accounted + and right to left find graphically the result of the following operations, and verify by calculation : (a) 2-36 + 3-01 -2-63 +1-06 3. (t>) 4-62 -1-68 +0-91 -3-42 + 2. (c) 4-62 -i -68 + 0-9 1 -3-42 -0-3. (d) i-87+o-53-o-35-i78. 0?) i -87 + 0-53 -0-36 -078. DIVISION OF DECIMALS 107 21. DIVISION OF DECIMALS. General Case. EXAMPLE. How many lengths each 23-28 cm. can be cut off a rod 345-681 cm. long? What also will be left over? The rod has to be divided, and the problem is the same as if we were told that each length was 2-328 dm. and the rod 34-5681 dm. or that each length was 232-8 mm. and the rod 3456-81 mm. That is to say, the decimal point in the divisor can be moved any number of places to right or to left, provided that in the dividend the decimal point be moved the same number of places to the right or to the left as the case may be. It will be most convenient to move the decimal point until, in the divisor, the digit of highest place value is the units digit. Thus 345-681 -7-23-28 = 34-568 1 -7-2-328, where the decimal point has been moved one place to the left in divisor and dividend, so that in the divisor 2-328 the figure of highest place value, i.e. the first 2, is in the units place. Remembering that the divisor is between 2 and 3, proceed as in short division until the first significant figure is obtained ; here it is easily seen to be i, its position is over the 3; and the first part of the quotient is 10. 4- = 2nd part of Quotient. io- = ist part of Quotient. 2-328 Proceed now as in ordinary division and obtain the 11-2881 ... first partial remainder ; i.e. 11-2881. 9-3 1 2 Divide again ; the 2nd part of the quotient is 4, its position is over the 4 ; and the 1-9761 ... remainder is 1-9761 dm. MMM Hence (10 + 4), i.e. 14 lengths, can be cut off, and a piece 1-9761 dm. or 19-761 cm. will be left oven io8 A MODERN ARITHMETIC The work may be shortened by writing remainders only, and then appears thus : 14 = Quotient 2-328 34.5681 11-2881 1-9761 = Remainder in dm. The problem of division may be expressed in another way, thus : What number must 23-23 be multiplied by so as to yield 30-299? In problems of this kind the divisor may be greater than the dividend, and the quotient will be a decimal less than unity. EXAMPLE. To find the value 0/0-323 -^62-5. As in the last example, the decimal point is moved in divisor and dividend, until in the divisor the digit of highest place value is in the units place ; 0-0323 -f- 6-25 is thus obtained. . Remembering that in any decimal, cyphers may be added to the right of the last decimal figure, without altering the value of the decimal, the procedure is as in the last example. 0-000008 4th part of the quotient. 0-00006 3rd o-oooi 2nd o-ooc; ist 6-25 O-O323OOOO 0.03125 0-OOI050 O-OOO625 0-0004250 0.0003750 0-00005000 O-OOOO5OOO ...ist remainder. 2nd remainder. , ...3rd remainder. The complete quotient is 0-005168. DIVISION OF DECIMALS 109 Many of the cyphers are often omitted in actual work, thus : 0-005 1 68 = Quotient 6-25 0-03230000 0-03125 1050 625 4250 375 5000 5000 or writing remainders only : 0-005168 Quotient. 6-25 | 0-03230000 1050 4250 5000 It will be noticed that when the necessary cyphers are added to the dividend, the number of decimal figures in the dividend is equal to the sum of the number in the divisor 'and the quotient. EXERCISES XXXII. a. Mental or Oral 1. Divide 4 by (a) 0-4, (b) 5, (c) 0-8, (d) 20. 2. 16 by (a) 5, (b} 50, (c) 0-8, (d) 0-08. 3. 24 by (a) 8, (b) 0-6, (c) 60, (d) 0-3, (e) 300, (/) 40. 4. 240 by (a) 6, (b) 60, (c} 600, (d} 0-6, (e) 0-12, (/) 800. 5. 30 by (a) 4, (6) 40, (c) 75> (d) 5> (e} 0-5, (/) 0-6, (g) 600. 6. 48 by (a) 8, (b} 80, (c) 60, (d} 12, (e} 0-12, (/) 2-4, Or) 24, (/) 240. 7. 4-8 by (a) 0-3, (b) 0-48, (*) 0-6, (</) 8, (e) 0-8, (/) 16. HO A MODERN ARITHMETIC 8. Divide 7-2 by (a) 2-4, (6) 72, (c) 0-9, (d) 0-72, (e) 0-8, (/) 0-24, Or) 36, (A) 3-6. 9. 0-64, by (a) 0-4, (b) 6-4, (<:) 8, (rf) 0-8, (e) 32, (/) 0-32, Or) 1-6. 10. 9-6 by (a) 12, 0) 0-8, (c) 0-16, (rf) 3-2, (*) 48, (/) 480, () 0-6, (X) 0-06. 11. How often is 0-25 of a yard contained in (a) 5 yds., (b) 50 yds., (<:) 4 yds. ? 12. How many times is 0-8 of a Ib. contained in (a) 5 lb., (b) 7-2 lb., (<:) 1 6 lb. ? 13. Divide 7-5^. into (a) 5^., (3) 2j. 6^., (<:) IQJ. and (d) i. 14. By what must 3-2 be multiplied to give (a) 1-6, (b) 48 and (c) 80 ? 22. DIVISION TRUE TO A CERTAIN NUMBER OF DECIMAL PLACES. Just as in earlier problems (p. 36), results were required to the nearest unit or to the nearest hundred, etc., so frequently in the division of decimals the process need not be carried on very far ; it is enough to obtain the result true to a certain number of decimal figures. Thus, in dividing 0-0423 by 7, we obtain 0-006,04285... . The result of the division is 0-006 true to three decimal places, 0-006043 true to tne nearest figure in the sixth place, and so on. EXERCISES XXXII. b. 1. The weight of one cubic foot of iron being 485 lb., and the weight of one cubic foot of water being 62-5 lb., how many times heavier is iron than water? (In other words, find the specific gravity of iron.) DIVISION OF DECIMALS ill 2. What is the volume in cubic inches (true to 3 decimal places) of a 7 Ib. cast-iron weight when a cubic inch of it weighs 0-26 Ib. 3. Given that a metre is equal to 39-37 in., what multipliers must be used to convert metres into yards and to convert yards into metres (true to four places of decimals)? 4. A litre is equivalent to 1-7617 pints; knowing that 8 pints make I gallon, express 36 gallons in litres (true to two places of decimals). 5. How many times can a vessel holding 3-45 pints be filled from one holding 18-732 gallons, and what amount remains? (i gallon equals 8 pints.) 6. A metre being known to be equal to 39-37 inches ; how many metre rods could be cut off from a rod 10 yards long? What would be the length of the piece left over ? 7. How many metres could be cut off from a rod 7 ft. long ? State the length of the piece remaining. 8. If one cubic foot of brass weigh 551-25 Ib., how many cubic feet will there be in a brass block weighing 1290-95 Ib. ? 9. When a certain screw is turned round through one complete revolution, its head goes forward by 0-132 of an inch. How many turns must you give such a screw to drive it into a piece of wood to a depth of 1-716 inches? 10. The pitch or a screw being 0-166 inch, how many turns must you give it to move it forward through a distance of 1-328 inches ? (The pitch of a screw is the distance the screw moves forward when turned through one complete turn or revolution.) 11. Divide (a) 108-1 by 2-3, then write down the quotients of (b) 10-8 1 by 0-23, (c) i -08 1 by 230, (d) 001081 by 23, (e) 1-081 by 0-0023. 12. Divide (a) 2-511 by 0-27, then write down the quotients of (b) 251-1 by 27, (c) 0-002511 by 0-027, (d) 0-2511 by 2700, (e) 2511 by 0-0027. 13. Divide (a) 3024 by 3-6, then write the quotients of (b) 3-024 by 3'6, (c) 30-24 by 0-036, (d) 0-03024 by 360, (e) 0-003024 by 0-0036. U2 A MODERN ARITHMETIC 14. Divide (a) 336-42 by 0-54, then write the quotients of (b) 0-33642 by 540, (c) 0-033642 by 5-4, (d) 3-3642 "by 0-0054, (e) 3-3642 by 54. 15. Divide (a) 500-208 by 0-68, then write the quotients of (b) 5002-08 by 0-68, (^r) 5002080 by 0-068, (d) 0-00500208 by 680, (e) 0-500208 by 6800. 16. Divide (a) 5726-885 by 60-283, then write the quotients of (6) 57-26885 by 602-83, (c) 572-6885 by 6028-3, (d) 0-5726885 by 0-060283, (e) 0-05726885 by 60-283. 17. Divide (a) 29019-69 by 36-5, then give the quotients of (b) 290-1969 by 0-365, (c) 0-02901969 by 365, (d) 2901-969 by 3650. 18. Divide (a) 74-820025 by 1867, then give the quotients of (b) 0-74820025 by 18-67, (c) 7482-0025 by 18670, (d) 748200-25 by 186-7, (?) 7-4820025 by 0-01867. 19. Divide (a) 5942-6842 by 73-4, then give the quotients of (b) 59-426842 by 734, (c) 59426-842 by 7340, (d) 0-59426842 by 0-0734, (e) 594-26842 by 0-00734. 20. Divide (a) 281-37084 by 57-9, then give the quotients of (d) 28-137084 by 579, 0) 281370-84 by 57900, (d) 0-28137084 by 5-79, (e) 2813-7084 by 0-0579. 21. Divide (a) 117-1408 bv 2091-8, then give the quotients of (b) 117140-8 by 20-918, (c) 1171-408 by 20918, (d) 1-171408 by 0-0020918. 22. Divide (a) 2791-490 by 3-94, then give the quotients of (b) 27-9149 by 394, (c) 0-279149 by 39-4, (d) 27914-9 by 0-00394. 23. Divide (a) 71656-228 by 95-06, then give the quotients of (b) 71-656228 by 950-6, (c) 716562-28 by 9506, (d) 7165-6228 by 0-9506, (e) 716-56228 by 0-009506. 24. Divide (a) 240493-344 by 62-4, then give the quotients of () 2404-93344 by 6240, (c) 24-0493344 by 0-624, (d) 0-240493344 by 6-24, (e) 240493344 by 62400. 25. Divide (a) 882-54019 by 9070-3, then give the quotients of (b} 88254-019 by 90-703, (c) 8825-4019 by 9-0703, (d) 88254-019 by 907-03, (e) 88254019 by 9070300. BRITISH MEASURES OF LENGTH 113 23. BRITISH MEASURES OF LENGTH (continued}. Other British systems for measuring length exist, in addition to those mentioned already (p. 74). Thus, in surveying, we have 100 links = i chain, 25 links = i pole, 10 chains = i furlong, 8 furlongs = i mile. Some other measures used in special branches of work are the following : T Geographical mile or Nautical mile =1-15 ordinary or Statute mile, League =3 'miles, Cable = about 100 fathoms, but a varying length, Fathom = 6 feet, Pace [Military] = 2 ft. 6 in., Pace [Geometrical] = 5 ft., Ell =1-25 yards, Cubit = 1 8 inches, Quarter or Span = 9 inches, Hand = 4 inches, Palm = 3 inches, Nail =2-25 inches, 12 Lines = 7 2 points =i inch. EXAMPLE. To express 2 yards 2 feet 9 inches in feet and decimals of a foot. Since 12 inches make i foot, divide 9 by 12 to change the inches to feet, .". 2 yds. 2 ft. 9 in. = 2 yds. 2-75 ft. = 8-75 ft. J.M.A. ii 4 A MODERN ARITHMETIC EXAMPLE. To express 3 miles 4 fur. 3 ch. 5 Iks. as a decimal of a mile. 3 miles 4 fur. 3 ch. 5 Iks, = 3 miles 4 fur. 3-05 ch. Since 100 Iks. = i ch. ) i Ik. =0-01 ch. J = 3 miles 4-305 fur. Since 10 ch. = T furlong, divide the number of chains by 10 to express in furlongs. = 3-538125 miles. Divide the furlongs by 8 to express in miles. EXAMPLE. To express 5-1627 miles in miles, furlongs, chains and links. 5-1627 miles Multiply the decimal part by 8 to express in furlongs. = 5 miles 1-3016 furlongs = 5 miles i fur. 3-016 chains Multiply -3016 by 10 to express in chains. = 5 miles i fur. 3 ch. 1-6 Iks. Multiply -016 by 100 to express in Iks. EXEECISES XXXIII Express as decimals of a mile : fur. ch. Iks. fur. ch. Iks. 1. 2 . 3 . 2. 2. 3 . 8 . 32. 3. 7 . 6 . 8. 4. 4 . 2 . 64. 5. 3 3 - 3- 6. 6 . 9 . 72. 7. 3 . 2 . 6. 8. 6 . 7 . 76. Express in furlongs, chains and links : 9. 3-26 of a mile. 10. 0-3284 of a mile. 11. 1-062 12. 0-089 13. 2-163 14. 4-361 15. 7-125 16. 0-375 17. 0-002 18. 6-8 1 19. A ship sails at the rate of 12 knots per hour ; how long will it take to complete a voyage of 1700 statute miles? (i knot or BRITISH MEASURES OF LENGTH 115 nautical mile =1-15 statute miles.) State the answer true to the nearest hour. NOTE. Much confusion exists relatively to the use of the term 'knot.' Some authorities treat it as in Ex. 19, i.e. as a distance. Other authorities use it as a speed, i.e. one knot is regarded as one geographical mile per hour. 20. A ship covers a distance of 288 miles in a day (24 hours). What is its speed in miles per hour ? 21. An express train ran a distance of 457-86 miles in 7-8 hrs. What was the average speed in miles per hour ? 22. Express in yards and decimals of a yard a distance of 7 ch. 29 Iks. 23. The distance between the wickets on a cricket pitch is i chain. Express in miles the distance a man runs in scoring 120 runs, 64 of which were run out, assuming the distance covered for each run to be the same as the distance between the wickets ? 24. Express in feet and inches the heights of the horses mentioned in the following extracts of advertisements : (1) Good Brougham Horse, 16-1 hands high. (2) Carriage Gelding, 15-2 (3) Black Brown Gelding, 16 (4) Bay Cob Gelding, 15 (5) Bay Gelding, 15-8 (6) High stepping Bay, 14-3 (7) Cleveland Bay, 15-3 25. In the given table of velocities, some of which are only approximate, fill in the last column : Centimetres Feet Kilometres Miles Miles per per sec. per sec. per sec. per sec. hour. (i) Light 3xio 6 984 xio 6 300000 186420 ... (ii) Submarine Cable Signal 4 x io 8 31-1 x io 6 4000 2486 (iii) Earth in its orbit 30 x io 5 98-4 x io 3 30 18-6 ... (IV) P 4t ' he Ear ' h ' S } 465 *.o* -5*5* 0.465 (v) Sound in the atmo-l sphere und conditions sphere under average j- 34-96 x io 3 1147 0-3496 0-2225 ii6 A MODERN ARITHMETIC 26. Using the table of velocities in Exercise 25, find how long light takes in coming from the sun to the earth, an average dis- tance of 93-3 x io 6 miles. (State your answer true to the nearest second.) 27. The flash of a gun is noted by an observer, and 21-1 seconds afterwards he hears the report ; how far off is the gun from the observer? State your answer true to the nearest decametre. (Use the table of velocities in Exercise 25.) 24. AREA. The area of a square, of which the side is one foot long, is called one square foot (generally abbreviated into i sq. ft.). If the side of the square is an inch, the area is a square inch ; if the side is a centimetre, the area is called a square centimetre, and so on. EXERCISES XXXIV. Graphic. \Apparatus : Squared Paper. Dividers. Scales.} Construct on squared paper the following rectangles and find their area : 1. Length 4 in., breadth i in. 2. Length 8 cm., breadth i cm. 3. Length 3 in., breadth 2 in. 4. Length 82 mm., breadth 22 mm. 5. Length 3 cm., breadth 4 cm. 6. Length 5 in., breadth 3 in. 7. Length 6-2 cm., breadth 5 cm. 8. Length 12 cm., breadth io cm. 9. Length 6-3 cm., breadth 4-4 cm. 10. If the length be a cm. and the breadth represented by b cm., how would you express the area ? AREA 117 Having seen that the number of units of area contained in a rectangle is equal to the product of the number of units oj length and the number of units of breadth, the ' tables ' of area can be easily understood. 25. BRITISH MEASURES OF AREA. (12 x 12), i.e. 144 square inches make i square foot, (3 X 3)> ** 9 square feet i square yard. In surveying, in the measurement of fields, and for similar purposes, the following system is in use : (loox 100), i.e. (10,000) square links make i square chain, (4x4), i.e. 1 6 square poles i square chain, (22 x 22), i.e. 484 square yards i square chain, 1 60 sq. po. or 10 square chains an area known as i acre (represented generally as i ac.). The fourth part of an acre is known as the rood (abbreviated into ro.). Accordingly, i ac. = 4 ro. = 1 60 sq. po. = 10 sq. ch. = 4840 sq. yards = 100000 sq. links. Since i mile contains 80 chains, i sq. mile = 6400 sq. ch. = 640 ac. NOTE ON TABLE. The acre was the amount of land which could, on a rough estimate, be ploughed by a horse in a day. It was generally regarded as a space 220 yds. long (one furlong, i.e. one furrow long) ; the breadth was a chain, it was divided into 72 furrows. Units of Area. 144 sq. in. = i sq. ft, 9 sq. ft. = i sq. yard, i oooo sq. links = 484 sq. yds. = 16 sq. po. = i sq. chain, 4840 sq. yds. = 100000 sq. links = 160 sq. po. = 10 sq. ch. = 4 ro. = i acre, 640 acres = i sq. mile. ii8 A MODERN ARITHMETIC 26. METRIC MEASURES OF AREA. (IQX 10), i.e. 100 sq. mm. = i sq. cm., (10 x 10), i.e. 100 sq. cm. = i sq. dm., (100 x 100), i.e. i oooo sq. cm. = 100 sq. dm. = i sq. metre = i centiare, 100 sq. metres = i sq. decametre = i are, 100 sq. decametres =i sq. hectometre =i Hectare, 100 sq. hectometres = i sq. kilometre. The terms centiare, are, hectare are used principally in dealing with area of fields, etc., and in surveying generally. EXAMPLE. Find the area of a rectangular field, length 8 ch. 3 Iks., breadth 4 ch. 25 Iks., expressing the answer in acres, roods, sq. po., etc. It is convenient to express both the length and the breadth in chains and decimals of a chain, thus : Length =8-03 ch. 8-03 Breadth = 4-25 ch. 4-25 Area =8-03 x 4-25 sq. ch. ^ 2 . 12 1-60 6 40 15 = 34-1275 sq. ch. 34-J2 75 = 3-41275 ac Divide by 10 to reduce to acres. = 3ac. 1-651 ro Multiply the decimal part by 4 to reduce to roods. = 3ac. i ro. 26-04 Multiply the decimal part sq. po. of the rood by 40 to reduce to sq. poles. .*. the area of the field is 3 acres i rood 26-04 square poles. METRIC MEASURES OF AREA 119 EXAMPLE. What is the length of a rectangular field, the area of which is 74 acres 6-806 sq. chains and the breadth of which is 20 ch. 63 links ? The number of sq. ch, in the area = (number of ch. in length) x (number of ch. in breadth). .*. number of chains in the length = (number of sq. ch. in area) -r- (number of ch. in breadth). Area = 74 ac. 6-806 sq. ch. = 746-806 sq. ch. 36-2 Breadth = 20 ch. 63 links =20-63 cn - 2-063' 74'68o6 Length = 746-806-=- 20-63 6189 36-2 chains 36 ch. 20 Iks. 12 790 12378 4126 4126 EXAMPLE. A room is 12 ft. 8 in. long, 1 1 //. 4 in. broad. How many blocks of wood (parquets] each 8 inches square will be required to floor it ? Area= 152 x 136 sq. inches 152 136 152 45 6 912 = 20672 sq. inches. 20672 Area covered by i parquet = 8 x 8 = 64 sq. in. Number of parquets required = 20672 -e- 64 = 323. EXAMPLE. A field is 150-3 metres long, 89-7 metres broad. Find its area, and express in ares. Area of field = 150-3 x 89-7 sq. metres I 5o*3 = 13481-91 sq. metres 8 9-7 12024. I352-7 = 134 ares 81-91 sq. m. (dividing by 100 ) *5' 2 I 13481-9 i 120 A MODERN ARITHMETIC EXAMPLE. How many acres, roods, etc., are there In a field having an area 0-32625 of a sq. mile? 0-32625 0-32625 sq. mile 640 = 0-32625 x 640 ac. 195-750 = 2o8 ' 8ac - 13-0500 = 208 ac. 3-2 ro. 208 ac. 3 ro. 8 sq. po. 208-8 EXERCISES XXXV. a. 1. The floor of a room is 32 sq. yds. 4 sq. ft. 36 sq. in. ; part of it is covered by a 'square' of carpet 20 sq. yds. 8 sq. ft. loo sq. in. , find the area of the portion of the floor not covered by the carpet. 2. The floor of a room is partly covered by a square of carpet 14 ft. long 12 ft. broad, the area of the remaining portion is 8 sq. yds. 3 sq. ft. What is the total area of the room ? 3. A block of wood is 8-34 cm. by 6-21 cm., in section ; through it is bored a hole 20-3 sq. cm. in extent. What does the area of *he section now become ? 4. The section of a girder is in the form of a capital H, where the two vertical strokes are 14" by 2-2", the cross portion 8" by 1-2". What is the total area of the section ? 5. Three rooms communicate by folding doors. One room is 16 ft. long 12 ft. broad, the second is 13 ft. long 12 ft. broad and the last is 16 ft. long 15 ft. broad. Express the area of each room in sq. yds. and sq. ft., and find the area of the carpet necessary to cover the three. 6. Find the sum of the following areas : sq. yds. sq. ft. sq. in. 13 . 3 . 68 14 . 8 . 75 9 . 7 . 100 1.6. 37 METRIC MEASURES OF AREA 121 7. The tiled hearth of a fireplace is covered with 98 tiles, each tile being square side 2 inches. What is the area of the hearth ? Express in sq. ft. and sq. in. 8. What is the area of a floor covered with 1200 parquets each 9 in. long 4 in. broad ? Express your answer in sq. ft. 9. What is the area of a yard paved with 84 flagstones each 2 ft. 4 in. long, and I ft. 10 in. broad ? 10. The top floor of a house contains the following rooms : Bedroom - 17 ft. by 13 ft. - H ft- 9ft- - 12 ft. 8 ft. Box room - 1 3 ft. 6 ft. If the inside dimensions of the house on that floor be 23 ft. by 27 ft, what area is used for passages, landings and staircases ? 11. A square field is 70 yards long ; find its area in acres and decimals of an acre (true to 4 significant figures). [N.JB. Make a mental note from this, as to the approximate extent of an acre.] EXERCISES XXXV. b. Mental or Oral. State roughly the number of acres in the following fields, assuming an area of 70 yd. x 70 yd. to be approximately an acre. (i) 140 yd. by 70 yd. (ii) 280 yd. by 210 yd. (iii) 140 yd. 35 yd. (iv) 350 yd. square. (v) 630 yd. 280 yd. (vi) 105 yd. by 140 yd. (vii) 700 yd. 350 yd. (viii) 560 yd. 210 yd. (ix) 700 yd. 280 yd. (x) 245yd. 210 yd. 122 A MODERN ARITHMETIC The fields, the areas of which are given below, may be taken as approximately rectangular. One side of the field is given ; find approximately in yards the length of the other 1. Area= Sac., one side 70 yd. 2. = 40 ac., 280 yd. 3. = 36 ac., 630 yd. 4. = 56oac., 560 yd. 5. = 24 ac., 210 yd. 6. = loo ac., 1400 yd. 7. = 35 ac., 490 yd. 8. = 8 ac., 140 yd. 9. = 12 ac., 140 yd. 10. = 6 ac., 21 yd. 11. = 84 ac., 49 yd. 12. = i Sac., 350 yd. EXERCISES XXXV. c. Practical. \Apparatus : Squared Paper. Dividers. Scales. Rulers."] 1. Draw the plan of a room 1 5 ft. long, 13 ft. broad, representing each foot by a distance of i cm. ; divide it up into strips 2 ft. 6 in. wide ; show that this can be done exactly if the length of the strips runs in the direction of the breadth of the room. If you wished to cover the floor of the room with carpet, 30 inches wide, find the length of carpet required, in order that none be wasted. Show that if the carpet be arranged so that the width be in the direction of the width of the room, some carpet will be wasted ; find how many sq. ft. will be wasted, also the length of carpet required in this case. METRIC MEASURES OF AREA 123 2. Draw the plan of a room 18 ft. by 16 ft. (scale i cm. to the foot). Find the length of carpet required to cover its floor with carpet 2 ft. 3 in. wide arranged so that the wastage is least. 3. Draw the plan 01 a passage 20 ft. long, 6 ft. broad (scale 4 ft. represented by I inch). Show it covered with flagstones 2 ft. long, 2 ft. wide. How many are required ? 4. Draw a plan of a room 15 ft. by 13 ft. Find the amount of timber required to floor it, the timber being boards 9 in. wide, and arranged with their length in the direction of the width of the room (scale i foot represented by i cm.). 5. Draw a representation of a roof, 20 ft. by 15 ft. (scale 4 ft. represented by i inch). Show it covered with slates, each slate being i ft. long, and 9 in. wide. How many slates will be required, supposing no overlapping of one slate over another ? 6. Draw a representation of a roof 24 ft. long by 15 ft. broad, covered with slates i ft. by 9 in., each slate lapping over the one underneath it to a distance of 4-5 in., and the length of each slate parallel to the length of the roof. How many slates will be re- quired? (Represent a distance of 1-5 in. by i mm.) 7. Draw the roof in the preceding question, where the lap is 3 in. only, and find the number of slates required in this case. 8. Draw a plan of a room 16 ft. long, 12 ft. wide. Show it carpeted with carpet 2 ft. wide, the width of the carpet being parallel to the width of the room. Suppose the carpet has a pattern which recurs every 5 ft. ; mark the points of recurrence by small arrowheads, and find the amount of carpet wasted if, in the different strips, corresponding points have to be alongside of one another. 9. A room is 14 ft. long, 12 ft. broad, 1 1 ft. high. Cut out a piece of paper which, when folded round, would represent the four walls of the room, and find the total area of the wall (scale 4 ft. repre- sented by i cm.). 124 A MODERN ARITHMETIC 10. Draw the following rectangles ; show by cutting them into two, along the diagonal, superposing one portion on the other, that the diagonal divides the rectangle into two equal portions : (a) 12 cm. by 8 cm., (b) 3-62 in. by 2-41 in., (c) 5 in. by 3 in., (d) 15 cm. by 6 cm. 11. Draw a circle 3 in. radius ; take any point O within it, at a distance of I in. from the centre ; draw any three chords AOB, COD, EOF. Find the areas of the rectangles the sides of which are (a) OA and OB, (b) OC and OD, (c) OE and OF. 12. Draw a rectangle 62 mm. long, 1 2 mm. broad ; fold it so as to represent the four walls of a room 18 ft. long, 13 ft. broad and 12 ft. high (scale i ft. represented by i mm.). State the area of the four walls. Find the cost of painting at 6d. per sq. ft. 13. Draw a rectangle and fold it so as to represent the four walls of a room 15 ft. by 12 ft. and 10 ft. high, representing i ft. by i mm. What would be the cost of painting its four walls at 6d. per sq. ft. ? 14. Draw, cut out and fold a representation of the sides and base of a box i ft. long, i ft. broad and i ft. high (scale i ft. represented by i in.). What is the area of the inside surface ? 15. Draw, cut out and fold a representation of a box (sides and base) 2 ft. long, 6 in. broad and i ft. high ; find the area of its inner surface and the cost of lining it with sheet lead at gd. per sq. ft. (Represent a distance of i ft. by a line i in. long.) 16. Draw, cut out and fold paper so as to form a triangular prism (Fig. 16) 4 inches long, and the ends equilateral triangles of one inch side ; find its surface exclusive of the triangular ends. FIG. 16. A Triangular Prism. 17. Draw a representation of the four walls of a room 16 ft. long, 14 ft. broad, 10 ft. high ; mark on it a frieze of i ft. (that is, a band METRIC MEASURES OF AREA 125 i ft. wide from the ceiling round the room), also a window in the centre of one of the long side walls, a door at the end of that wall, and a fireplace in the opposite wall, the dimensions being : Window, 8 ft. high and 6 ft. broad. Door, 8 ft. high and 4 ft. broad. Fireplace, 5 ft. high and 3 ft. broad. (Represent i ft. by i mm., and calculate the remaining area of the walls.) 18. In Ex. 17 mark off paper strips 2 ft. wide suitable for paper- ing. What length of paper would be required ? 19. Draw, cut out and fold paper to represent a rectangular prism (Fig. 17) 2 ft. long, 1-5 ft. broad, and i ft. thick. (Represent I ft. by i inch.) If gilded at $s. per sq. ft, What would the COSt amount to ? FIG. i 7 .-A Rectangular Prism. EXERCISES XXXV. d. 1. Find the total surface of a brick 9 in. long, 4-5 in. broad, 3 in. thick. 2. What is the total surface of a square prism 10 in. long with an end of 2 in. edge ? 3. Find the inner surface of a box 4 ft. long, 3 ft. broad, 2 ft. deep. 4. What is the cost of painting the walls of a room 16 ft. long, 15 ft. broad, 11 ft. high, at is. ^d. per sq. ft., making no allowance for windows, doors, etc. 5. What length of paper 21 in. in width will be required to paper a room length 20 ft., breadth 15 ft., height 12 ft.? 6. If i metre is equal to 39-37 in., find (true to 4 figures) how many (a) sq. in., (b) sq. ft, (c) sq. yd. there are in a sq. metre. 126 A MODERN ARITHMETIC 7. Express a chain in metres, etc., to the nearest centimetre ; hence, find out how many square metres there are in an acre. State also the number of ares, etc. 8. A man buys a plot of ground of 3 ro. 28 sq. po. ; on it he intends building a house with a frontage of 50 ft. and a depth of 60 ft. What area will he have left for his garden ? 9. A yard 60 yds. square is surrounded by a paved walk 3 ft. in width. If the walk be paved with flags 18 in. square, how many flags are there ? 10. A room is 19 ft. long, 17 ft. broad. If the floor is to be carpeted so as to leave a margin round the room 15 in. wide, what area of carpet will be required, and what will be the extent of the surrounding margin ? 11. A room has an area of 21-84 s( l- metres, and its breadth is 4-16 metres ; what is its length ? 12. A ribbon has a surface of 47-25 sq. dm., and the width is 1-5 cm. ; what is the length of the ribbon ? 13. How many sq. ft. could be covered by a piece of carpet 27 in. wide and 25 yds. long? 14. A table is 5 ft. long and 2 ft. 6 in. broad ; the table cloth is 8 ft. long and 5 ft. broad. What area of cloth hangs over the side of the table ? 15. Find what length of plain paper, 21 inches wide, will be required to paper the rooms described below. Make no allowance for doors, windows, and so on. (a) Length, 20 ft., breadth, 14 ft. 6 in., height, loft. 6 in. (0) 1 8 ft. 9 in., 1 2 ft. 9 in., 9ft. 6 in. (c) i6ft. 6 in., lift. 6 in., 9 ft. 6 in. 16. The four walls of a room have a total area of 660 sq. ft. ; the area of the floor is 270 sq. ft. ; the width of the floor is 15 ft. What is the height of the room ? METRIC MEASURES OF AREA 127 Measurement of Irregular Areas. Irregular areas can be calculated by representing the areas to scale on squared paper and counting the number of squares covered by the figure. EXAMPLE. To determine the area of the county of Norfolk. Each small square, in Fig. 18, represents an area of 9 sq. miles. In estimating the portions of squares, parts less than half a square are neglected, and portions greater than half a square are counted as whole squares. FIG. 18. Map of Norfolk. (Scale 30 miles to the inch.) Thus, in estimating the area, we practically work as if the area was bounded by straight lines such as those shown in Fig. 1 8. On counting then the number of squares included in the county, we get 5 + 5 + 6 + 10+11 + 11 + 11 + 11 + 11 + 12 + 13+11 + 11+9 + 8 + 7 + 6 + 5 = 202, and the area of Norfolk is thus found to be 9 x 202 or 1818 sq. miles. EXERCISES XXXVI. Practical. Apparatus : Squared paper, tracing paper. Find the following areas : 1. Switzerland. 2. Hyde Park. 3. Cornwall. 4. Isle of Man. 5. Isle of Wight. 6. Wigtown. 7. Anglesea. 8. Arran. Scale, i square represents i square mile T.M.A. FIG 20 r 130 A MODERN ARITHMETIC 27. DECIMAL COINAGE. In the majority of the more important countries a decimal system of coinage is used. Thus : In France, Switzerland and Belgium, sums of money are expressed in francs and centimes, where i franc (fr.)=ioo centimes, the franc being equivalent to about 9^. of British money. In Italy francs are replaced by lires, centimes by centesimos ; and, in Greece, drachmas and leptas take the place of francs and centimes. In Germany i krone = 10 marks =1000 pfennigs. The mark is of nearly the same value as our shilling. In the United States i dollar =100 cents. The cent is about equal to the British \d. in value. Some approximate equivalents of foreign coinage in British money are shown in the accompanying table : Approximate Values of Foreign Coinage. England. France, Switzerland, Belgium. Italy. Greece. United States. Germany. . S. D. Frs. Cts. Lrs. Cts. Drs. Lpts. Dols. Cts. Mks. Pfgs. I O IO O IO O IO O 2 8 003 o 31 o 31 o 31 6 o 25 006 o 62 o 62 o 62 12 o 50 9& I I I O o 19 o 79 010 I 25 I 25 I 25 o 25 I 040 5 o 5 o ? o I 4- o O IO O 12 50 12 50 12 50 2 50 10 IOO 25 o 25 o 25 o c o 20 DECIMAL COINAGE 131 EXERCISES XXXVII. a. Mental or Oral. 1. Express as francs and centimes the following amounts : (a) 2320 centimes. (b) 19360 centimes. (c) 8370 centimes. (d) 9-36 francs. (e) 1945-3 francs. (/) 876-2 francs. (<") 897-60 francs. (h) 1-7623 francs. 2. Express (a) 6-3 lires in terms of lires and centesimos. () 783 centesimos (c} 9-3 dollars in terms of dollars and cents. (d) 3-756 dollars (e) 821 cents GO 3568o leptas in terms of drachmas and leptas. Of) 93 2I 3 pfennigs in terms of marks and pfennigs. (/&) 1-32 krones 3. Find the cost of (a) (i) 10 articles at 15 cents each, (ii) 20 articles at 13 cents each. (iii) loo articles at I dollar 3 cents each. (b) (i) loo articles at I franc 50 centimes each, (ii) 400 articles at 30 centimes each. (iii) icoo articles at 10 centimes each. (c} (i) 8 articles at i drachma 40 leptas each. (d) (i) 120 articles at 15 lires 20 centesimos each. (ii) 100 at 3 lires 20 centesimos each. (e) (i) 500 articles at i mark 3 pfennigs each, (ii) 25 at 4 marks 6 pfennigs each. EXERCISES XXXVII. b. 1. If 35 raetres of cloth cost 26 fr. 60 c., what, is the cost per metre ? 2. If 89 metres of cloth cost 418 fr. 30 c., what is the cost per metre ? I 3 2 A MODERN ARITHMETIC 3. What is the cost of 23 metres of silk at 5 fr. 70 c. per metre ? 4. What is the cost of 31 metres of cloth at 2 lires 90 centesimos per metre ? 5. Find the cost of 320 Ib. of metal at 96 cent s per Ib. 6. Find the cost of 81 kilo, of a substance costing 29 marks 6 pf. per kilo. 7. 3-05 kilo, of a substance cost 231 marks 8 pf. ; what is the cost per kilo. ? 8. Find the cost of 63 yd. of cloth worth 19 doll. 5 c. per yd. 9. Find the cost of 7560 kilo, of a substance at 6 fr, 40 c. per kilo. 10. 693 Ib. of ore are worth 9826 doll. 74 c. ; what is the value per Ib. ? 11. If 5150 fr. 80 c. be the cost of 15-8 kilo, of a substance, what is the cost per kilo.? 12. Find the cost of 1-912 kilo, of a substance at 53 lires 4 c. per gram. 13. Find the cost of 319 yds. of cloth at 10 doll. 35 c. per yd.? 14. 3194-5 metres of silk cost 24278 fr. 20 c. ; what is the cost per metre ? 15. 2299 krones i mark 40 pfennigs is the cost of 1446 kilo, of an ore; what is the cost per kilo.? 16. Find the value of 14-12 kilo, of copper at 5-43 krones per kilo. 17. Find the value per metre of cloth, when a piece 737 metres long is worth 5011 drachmas 60 leptas. 28. VOLUMES. The volume of a block, each edge of which is one inch long, is denned as a cubic inch; similarly the volume of a block, each edge of which is one centimetre long, is defined as a cubic centimetre (c.c.). VOLUMES 133 EXERCISES XXXVIII. a. Mental or Oral. 1. On each inch square, into which a rectangle 6" x 4" can be divided, a cube of one inch side is placed. What is the volume in cubic inches of the solid formed by the cubes ? What also are the dimensions length, breadth and thickness of the solid ? 2. Give the answers to Exercise I if the rectangle be replaced by a rectangle of sides : (a) 4" x 3", (#) 3" x 2", (c) 8" x 5", (d) 36" x 1 2", (e) 12" xi 2". 3. If on each cube in Exercises i and 2 another cube be placed, write down the dimensions and volume of the new solid formed. 4. If in Exercise 2 there be seven of the one cubic inch blocks, one above the other, placed on each square inch of the rectangle, what will be the dimensions and volume of the solid formed ? 5. Write down the volumes of the following blocks : (a) Length 8 in.,- breadth 6 in., thickness 4 in. (*) 6 in, 4 in. 3 in. (c) 11 12 metres, 8 metres, 6 metres. (d) 11 1 5 ft. 8ft, 11 3ft- w 11 12-5 in. 7-5 in., 11 4 in. (/) 11 25 cm, 18 cm. 11 14 cm. Ur) 11 x cm, y cm., 11 z cm. (h) 11 a inch, inch, 11 <: inch. 6. Find the number of cubic inches in a cubic foot, i.e. in a cube of 12 inches edge ; also the number of cubic feet in a cubic yard, i.e. in a cube of 3 feet edge. 7. Find the number of cubic centimetres in (a) a cubic decimetre, i.e. a volume equal to that of a cube of 10 cm. edge ; (b} a cubic metre. I 34 A MODERN ARITHMETIC TABLES OF VOLUME OR CUBIC MEASURE. British. i2 3 = 1728 cubic inches make i cubic foot, 3 3 = 27 cubic feet i cubic yard. Metric. io 3 = TOCO cubic millimetres make i cubic centimetre, io 3 = 1000 cubic centimetres ,, i cubic decimetre, io 3 = 1000 cubic decimetres ,, i cubic metre. A cubic decimetre is known as a Litre. A cubic metre is known as a Stere. EXERCISES XXXVIII. b. 1. A brick wall 54 ft. long, 10 ft. high and 9 in. thick is built of bricks the dimensions of which (inclusive of mortar) are length 9 in., breadth 4-5 in., thickness 3 in. Find the volume of each brick, also the number of bricks used in the construction of the wall. 2. A room is 18 ft. long, 10 ft. 6 in. high and 15 ft. broad ; what is its cubic content ? 3. A board is 4 m. long, 22 cm. broad and 2 cm. thick ; what is the volume of the board ? 4. A box has the following for its inside and outside dimensions (exclusive of cover) : Inside Length 3 ft. 3 in., Outside Length 3 ft. 5 in., breadth 2 ft. 4 in., breadth 2 ft. 6 in., depth 2 ft. depth 2 ft. i in. What is the volume of the wood forming the box ? VOLUMES 135 5. What is the thickness of a block of wood, the length of which is 4 ft., the breadth 3 ft. 6 in. and the cubic content 1 8 cub. ft. 1152 cub. in.? 6. A sheet of iron is 3-24 metres long, 2-58 metres broad, and its volume is 2089-8 cubic centimetres. What is the thickness of the sheet ? 7. A wooden box with its cover on, is an exact cube internally of i ft. edge ; what volume of lead 0-25" thick is required to line it ? 8. A room has a capacity of 3108 cubic feet ; its length is 18 feet 6 inches, its height is 12 feet ; what is its breadth? 9. A coverless box the internal dimensions of which are 4 ft. in length, 7 in. in breadth and 9 in. in depth, is made from 9 in. planks i in. in thickness ; what length of plank will be required, also what will be the volume of the planking ? 10. A swimming bath is 100 ft. long and 36 ft. broad ; when a number of men dive into the bath, the height of the water rises by half an inch. If the average amount of water displaced by one of the men is 2 cub. ft, how many men are there in the bath ? 11. A biscuit tin has internal dimensions as follows : Length 30 cm., breadth 30 cm., height 25 cm. It is partially filled with water and the depth of the water noted ; a glass jar is then pressed into the tin, until the water is just level with the top of the jar ; the rise of water in the tin is then found to be 3-2 cm. ; on pressing the jar further down, so that the jar becomes full and totally immersed, the level falls by 2 cm. Find the capacity of the jar, also the volume of the glass. 12. A plasterer has 1-26 cubic feet of cement in his bucket ; how many square yards of wall can be plastered to a thickness of 0-5 inch with it ? 13. The sluice supplying water from a mill pond to a mill wheel is 4 ft. 4 in. broad and 8 in. high. If the water rushes through it at the rate of 5-5 ft. per second, how many cubic feet pass through in one hour ? 136 A MODERN ARITHMETIC 14. A pontoon is 100 ft. long, 60 ft. broad and 30 ft. deep ; it is floated into a convenient position and filled with water ; how much water does it contain? If each cubic ft. of water weighs 62-5 Ibs., how much lighter does the pontoon become when the water is pumped out of it ? 15. A cistern is 5' long by 4' broad by 2' high, and is full of water. What will be the fall in the level of the water in the cistern if i quart is run out of it, assuming that 25 quarts make approxi- mately i cubic foot ? 16. A room is 15 ft. long, 12 ft. broad and 10 ft. 6 in. high. What is its content in cubic yards ? 17. A portion of a seam of coal is 240 yards long, 126 yds. broad and 4 ft. 6 in. in thickness. What weight of coal could be obtained from it, assuming that, roughly, each cubic yd. in the seam yields i ton at the surface ? 18. A stone cross is made from a slab of stone of which the section is a square of 6 in. side, the height of the cross is 5 ft, the breadth or span 3 ft. What is the total volume of the cross ? MEASUREMENTS OF CAPACITY. Many systems for measuring liquids exist for example : In measuring ordinary liquids,* 4 gills = i pint, 2 pints = i quart, 4 quarts = i gallon. In medicine, 60 minims (m.) = i drachm [fj], 8 drachms = i fluid ounce [f |, 20 fluid ounces = i pint. * More complete tables are given at p. 138. VOLUMES 137 EXERCISES* XXXIX. a. Practical. {Apparatus: Burette. Medicine bottle. Various rectangular tin boxes. Glass tumbler. Metal blocks. Scales. Medicine measure graduated. Pint mug.~\ 1. Measure in inches the length, breadth and depth of the given tin box ; calculate its cubic content in cubic inches (allowing for the thickness of the material of which the box is made) ; find by a burette the number of cubic centimetres the box holds. Hence calculate the number of cubic centimetres equivalent to i cubic inch, also the value of I c.c. as a decimal of a cubic inch. 2. Repeat Experiment 1 with other tin boxes. 3. Measure in inches the length, breadth and thickness of a given metal block and calculate its volume. Tie a fine piece of strong thread to the block and hang it in a glass tumbler, pom- water into the tumbler up to a convenient height (this must be sufficient to cover the block). Withdraw the block, removing as little water as possible. Fill the glass to the same height as before by the burette, noting the volume in c.c. of the water run into the tumbler. You now have the volume of the block in cubic inches and in cubic centimetres. Calculate the number of cubic centimetres in a cubic inch, also the number of cubic inches in a cubic centimetre. 4. Measure the dimensions of a given block in cm. and in inches, calculate the volume in c.c. and in cubic inches, and find the number of cubic centimetres in one cubic inch. 5. Repeat Experiment 3, using however a graduated medicine measure instead of a burette graduated in c.c. Hence find the number of fluid ounces in a cubic inch, also the number of cubic inches in a fluid ounce. *NoTE. IF may be convenient for many of these experiments to be performed by the teacher only. 138 A MODERN ARITHMETIC 6. How many fluid ozs. will fill a (i) tea spoon, (ii) dessert spoon, (iii) table spoon ? 7. Find the number of cubic centimetres in the given half-pint measure. The table stated on page 136 may be extended. Some of the more common measures are the following : LIQUID MEASURES. 4 Gills =i Pint (Pt). 2 Pints = i Quart (Qt). 4 Quarts = i Gallon (Gal.). 4^ Gallons = i Pin. 9 Gallons = i Firkin. 36 Gallons = i Barrel. 54 Gallons = i Hogshead. The gill, pint, quart and gallon are the only measures which can be regarded as standard, the others varying in different cases according to the custom of the particular trade using the measure. Thus i Hogshead of Ale is 54 gallons. I Claret,, 46 I Port 57 DRY MEASURE. Employed in measuring grain and corn. 2 Pints = Quart. 2 Quarts = Pottle. 4 Quarts = Gallon. 2 Gallons = Peck. 4 Pecks = Bushel. 8 Bushels = Quarter 5 Quarters = Load. 2 Loads = Last. Many kinds of cereals are however usually sold by weight, though nominally in bushels. PROBLEMS ON MEASURES OF CAPACITY 139 29. PROBLEMS ON MEASURES OF CAPACITY. EXAMPLE. One vessel contains 3 gallons i quart i pint ; another vessel contains 4 gallons 3 quarts i pint. What is the total content of the two vessels ? The process of addition is exactly similar to that employed in the addition of money and addition of length. 23,1. Qt. pt. 3.1.1 Adding the pints, we obtain 2 pts. or i quart. 4.3.1 Adding the quarts, we obtain (1+3 + 1) qt., - i.e. 5 qts. or i gal and i qt. Adding the gallons, we obtain 8 gal. i i The content is 8 gal. i qt. EXAMPLE. Express 16 gal. 3 qt. i pt. as gallons and decimals of a gallon. 1 6 gal. 3 qt. i pt. Divide the number of pints by 2, = 1 6 gal. 3.5 qt. to express them in quarts. = 4-875 gal. Divide the number of quarts by 4, to express them in gallons. EXAMPLE. A cask contains 53 gal. i qt. i pt. of ale. If this be divided equally among seven people, what does each person receive ? Here we have only to divide 53 gal. i qt. i pt. by 7. The method again is similar to that used in Division of Money. 53 gal divided by 7 yields 7 gal. and 4 gal. qt. pt. g al over. ' I DJ ' 4 gal. and i qt. = 1 7 qt. ; dividing by 7.2.1 Ans. 7, we get 2 qt. and 3 qt. over. 3 qts and i pt. = 7 pt. ; dividing by 7, we obtain i pt. exactly. I 4 o A MODERN ARITHMETIC EXERCISES XXXIX. b. 1. A gallon of wine will fill six ordinary wine bottles ; how many bottles of wine could be filled from a vat of wine containing 6 hogsheads 4 gal. 2 qts. ? What would be left over, if any ? (The hogshead of wine is equal to 57 gallons.) 2. Four and a half quarts of ale are run out of a nine-gallon cask ; how much ale remains in the cask ? 3. If a gill contains 8-665 cubic inches, how many cubic feet and cubic inches are there in 6-25 gallons ? N.B. It will be seen from this exercise that 25 gallons are very nearly the same as 4 cubic feet. 4. A cistern is 3 ft. long, 2 ft. broad and contains 125 gallons of water ; what is the depth of the water within the cistern ? (Assume i cubic foot = 6-25 gallons approximately.) 5. A decanter for holding spirits has for its inside section a square of 3 inch side ; a bottleful of whisky is poured into it. What is the height of the surface of the whisky above the bottom of the decanter ? (Six bottles to the gallon.) 6. If on a certain day the rainfall is 0-8 of an inch, how many gallons of water will fall on a field of 4 acres ? (Assume I cub. ft. = 6-25 galls.) 7. What is the volume of a cube of 3-9 inches edge ? Since a litre or cubic decimetre is the volume of a cube of 3-937 inches, the answer will be a little less than a litre. 8. Assuming that i litre = 61 cubic inches, how many cubic inches will there be in a rectangle 4 cm. long, 5-6 cm. broad, 3-7 cm. deep ? N.B. More accurately, i litre =6 1-027 cubic inches. PROBLEMS ON MEASURES OF CAPACITY' 141 9. If a litre is 61-02 cubic inches and I cubic foot is 6-25 gallons, how many pint mugs could be filled from a vessel containing 4 litres? How much would remain over ? Note therefore that approximately 7 pints = 4 litres. 10. In making up a medicine, 10 fluid ounces (f ^) 4 drachms (f 3) 24 minims (m.) of one liquid are added to 15 fluid ounces (f^) 7 drachms (fj) 49 minims (m.) of a second liquid. How much liquid is there altogether ? 11. Express 0-8645 quart in pints, fluid ounces, drachms and minims. 12. Add together the following : 7 f I 3 f 3 3 m - 5 8 f % 4 f 3 5 m - ; I2 f 3 3 f?> 25 m. 13. A medicine is to be made up of 14 f ^ 6 f 7) of liquid A, 8 f i 3 f 7) of liquid B, 12 f 4 f~> of liquid C. What would be the equivalent amounts to be taken by a dispenser who only had measures graduated according to the metric system ? 14. The internal section of a medicine bottle is a rectangle 2 inches long and I inch broad ; what is the distance in inches between the ounce marks on the bottle ? 30. MEASUREMENT OF WEIGHT. IN England many systems of measurement of weight are in use. Avoirdupois weight is the most general. Troy weight is used in dealing with gold and silver. Apothecaries' weight is used in dispensing, medicines. On the Continent the system is that in which the unit is the weight of -i cubic centimetre of water at a temperature of 4 C. and is called a gram ; the system is a decimal one. I 4 2 A MODERN ARITHMETIC The weights employed in the different systems are as follows : AVOIRDUPOIS WEIGHT. 1 6 drams = i ounce (oz.). 1 6 oz. =i pound (lb.) 14 Ib. = r stone (st). 28 Ib. = 2 st. =i quarter (qr.). 1 1 2 Ib. = 4 qr. = i hundredweight (cwt.X 2240 Ib. = 20 cwt. = i ton. TROY WEIGHT. 24 grains = I pennyweight (dwt.). 480 gr. = 20 dwt. = i ounce (oz.). 5760 gr. = 12 oz. =i pound. 100 Ib. = i hundredweight. (7000 grains troy = i Ib. avoirdupois.) APOTHECARIES' WEIGHT. 20 grains. = i scruple ()). 39=1 drachm (3). 85 =i ounce (5). \These are practically 5760 gr. = 12 =i pound (Ib.). / obsolete. N.B. The grain is the same weight in all three systems. METRIC SYSTEM. 1000 milligrams = zoo centigrams =i decigram =^i gram (gr.). 1000 grams = i kilogram (kilo.). The gram is the weight of one cubic centimetre of water at 4 C. In the Addition, Subtraction, Multiplication, Division and Reduction of weights the procedure is exactly similar to that employed in dealing with questions on money, length, area and volume. MEASUREMENT OF WEIGHT 143 EXAMPLE. How many pennies, three of which weigh i oz. avoir., can be made from a mass of 2 tons 3 cwt. i qr. 4 Ib. of Bronze ? tons. cwt. qr. st. Ib. oz. 231040 20 Reduce the 2 tons to cwt. by multiplying by 20 ; add on the * cwt. 43 4 Multiply by 4 to reduce from cwt. to qr. and add the i qr. 2 Multiply by 2 to reduce from qr. ,to st. 346 14 Multiply by 14 to reduce from st. to Ib., and add 6 the 4 Ib. 1384 4 4848 1 6 Multiply by 16 to reduce from Ib. to oz. 4848 29088 77568 77568 oz. = 232704 pennies. Multiply by 3, since 3 pennies weigh i oz. EXERCISES XL. a. Practical. [Apparatus : Balance, pennies, halfpennies, tumblers, quart mugs, medicine bottles, tin biscuit boxes, pieces of wire of 'various lengths, pieces of sheet qietal, cubes of metal of one inch side.] 1. Find the weight in ounces, of i penny, 4 pennies, 4 half- pennies, 8 pennies. 2. Weigh the given objects. I 4 4 A MODERN ARITHMETIC 3. Weigh the given half-pint glass, first when empty, secondly when full of water. Calculate from your results the weight of a gallon of water. 4. As in (3), find the weight of water in the given quart mug, and again calculate the weight of a gallon of water. 5. Weigh the given bottle and calculate its capacity, assuming from the result of (3) and (4) the weight of a gallon of water. 6. Find the weight of the given medicine bottle (i) empty ; (ii) containing (a) 2 f 5, (b) 4 f5, (c) 6 f of water. 7. Find the internal volumes in cubic inches of the given rectangular tin boxes, also the weights of water they contain when full. Calculate from your results the weight of a cubic foot of water. 8. Assuming the weight of a cubic foot of water, find the capacity of each of the given bottles. 9. Find the number of ounces avoirdupois in the given pound weight troy. 10. Find the weight of a litre of water in (a) grams, (b} ounces. 11. Express the weight of i Ib. avoirdupois in the decimal system. 12. Find by means of the balance the volumes in c.c. of the given vessel. 13. Express in ounces avoirdupois the given kilogram weight. 14. You are given a coil of wire, also one foot length of it ; weigh the two and calculate the length of the wire in the given coil. 15. You are given a rectangular sheet of metal one square inch in area, also an irregularly shaped piece of the same material and thickness ; find the weight of both and calculate the area of the given irregularly shaped piece. 16. Weigh the given solid, also the given inch cube of the same material ; from your results calculate the volume of the given solid. MEASUREMENT OF WEIGHT EXERCISES XL. b. 1. A train consists of eight trucks of coal, the net weights being respectively: 8 ton 3 cwt. 2 qr., 9 ton I qr., 7 ton 15 cwt. 3 qr., 8 ton 10 cwt. i qr., 8 ton 11 cwt. 2 qr., 8 ton 4 cwt, 9 ton i cwt. 3 qr., 7 ton 19 cwt. 2 qr. What is the total net weight of the train ? 2.1 The gross weight of a truck and load is 12 ton 3 cwt. 2 qr. ; the truck itself weighs 4 ton 10 cwt. 3 qr. What is the weight of the load the truck carries ? 3. A barrel of fish contains 3 qr. 15 Ib. 6 oz. ; if the barrel itself weighs 5 Ib. 12 oz., what is the weight of the fish? 4. Reduce to Ib. : (a) 4 cwt. 34 Ib., (b) 17 cwt. 3 qr., (c) 19 cwt. 45 Ib., (d) 14 ton 2 qr. 17 Ib., (e) 17 qr. 3 Ib., (/) 793 ton 14 cwt. 2 qr., (g) 19 ton 1041 Ib. Find also the sum of these weights. 5. Convert (a) 1345 oz. into Ib. (avoirdupois), (b) 7056 Ib. into cwt., (c) 19043 Ib. into tons, etc., (d) 1593 oz. into qr., (e) 793628 drams into cwt., (/) 6932 qr. into tons. 6. If i cubic foot of sea water weigh 64 Ib., what will be the weight in tons, cwt., qr., of 35 cubic feet? 7. What volume of sea water has a weight of 3000 tons 4 cwt? (i cubic foot of sea water weighs 64 Ib.) 8. When a body floats in water, it displaces a volume of water equal to its own weight ; what will be the volume of sea water displaced by a ship of 2$$o tons ? 9. What would be the volume of water displaced by the ship in Ex. 8, if it were in fresh water, and if i cubic foot of fresh water were supposed to weigh 1000 oz. ? 10. If a cubic foot of common brick work weigh i cwt., what is the weight of a brick wall 60 ft. long, 8 ft. 6 in. high and 9 in. thick ? J.M.A. K i 4 6 A MODERN ARITHMETIC 11. If a Welsh fire brick crushes when it is pressed with a force per square inch equal to the weight of 7 cwt. I qr., what is the greatest height a brick column could be built, without crushing under the action of its own weight ? (Assume each cubic inch of brick weighs 1-375 oz. and state the answer true to o-oi ft. 12. If i cubic foot of fresh water weigh 1000 oz., what will be the weight in tons, etc., of the water which fills a cistern 18 ft. long, 7 ft. 6 in. deep, 15 ft. 3 in. broad inside ? 13. A plate of iron is 64 cm. long, 57 cm. broad and weighs 11031-552 grams. If each cubic centimetre of iron weighs 7-2 grams, what is the thickness of the plate ? 14. How many kilogram weights could be made from 2 qr. 3 Ib. 4 oz. of brass ; what weight of brass will remain over ? (Assume i kilo. = 2-205 Ib.) 15. A litre of water weighs I kilogram and a gallon of water weighs 10 Ib. ; how many litre measures could be run out of a 9-gallon cask, and what will then be left in the cask ? 16. The wire on a reel weighs 13 cwt. 22 Ib. What is its length if the weight of a foot is known to be 5-6252 oz.? 17. Assuming that approximately i kilo, is equal to 2-2 Ib., express in French and English measures the following : (a) 6 tons 12 cwt., (b) 1000 kilo., (c) 4 cwt. 3 qr. 7 Ib., (d) 173 kilo. 500 grams, (e) 1-54 kilo. 18. A cart contains 1 5 sacks of flour each containing 2 cwt. 2 qr. 3 Ib. 2 oz. ; how much flour is there altogether in the cart ? 19. Evaluate the following : i ton 14 cwt. 3 qr. 10 Ib. 4 oz, multiplied by (a) 5, 0) 8, (c) 16, (d) 28, (e) 50, (/) 210, (g) 325. 20. A mass of iron weighing 37 tons 4 cwt. 3 qr. is divided into 24 equal parts ; what will each of these parts weigh ? 21. What is the weight per foot length of brass tubing such that a piece 124 ft. long weighs 5 cwt. 2 qr. n Ib. 12 oz.? 22. Divide 706 tons 3 cwt. 2 qr. 10 Ib. 4 oz. by (a) 8, (^) ir, (c) 35, (</.) 8 9; (e) 206, (/) 370. MEASUREMENT OF TIME 147 23. Express the following weights (avoirdupois) in Troy units, remembering that in i Ib. Troy there are 5760 grains, and 7000 grains in i Ib. avoirdupois : (a) 3 Ib. 2 oz., (b) 4 Ib. 6 oz. (c) I qr. o Ib. 8 c*., (d) 8 Ib. 4 oz., (e) i stone 10 oz., (/) 10 Ib. 12 oz. 24. Express the following weights (Troy) in avoirdupois measure. Express your answer in Ibs., oz., drs. and decimals of a dram (true to 2nd decimal place) : (a) 1 6 dwt. i gr., (b} i Ib. i oz. 14 dwt. 4 gr., (c) 19 Ib. 7 ot. i dwt. 1 6 gr., (d) 59 Ib. 7 oz. 5 dwt. 23 gr. 31. MEASUREMENT OF TIME. The chief divisions of time are : 60 seconds (60") = 60 minutes (60') = 24 hours 7 days 28 days or 4 weeks = 365 days minute, hour, day, week, lunar month, ordinary or civil year, 366 days = i leap year. EXERCISES XLI. 1. The earth makes a complete spin on its axis in 23 hr. 56 min. 4 sec. Find (a] the difference between this and the ordinary day, (b) what the difference amounts to in 365 days. 2. The time the earth takes in moving in its orbit once round the sun is i tropical y>sar or 365-2422164 days. Express this in days, hours, minutes and seconds. 3. Express as decimals of an ordinary year the following times : Spring - 92 days 20 hours 59 minutes. Summer - 93 14 13 Autumn - 89 18 35 Winter - 89 o ,, 2 State the answer to 4 decimal places. 148 A MODERN ARITHMETIC 4. Find the difference in hours and minutes between 3 ordinary years of 365 days together with i leap year ; and 4 tropical years. 5. What is the difference between 25 groups, each consisting of 3 ordinary and I leap year ; and 100 tropical years ? 6. If a lunation be 29-5306 days and a tropical year 365-242216 days, find the difference between 19 tropical years and 235 lunations. 7. Express a lunation in days, hours and minutes. 8. The following table gives (a) the periods of some of the planets about the sun, also (b) their distances from the sun in terms of the earth's distance : Planet. Mean distance from the Sun in terms of the Earth's distance. Period in Days. (a) Mercury - (b) Venus - (0 Earth - (et) Mars - (e) Jupiter - (/) Saturn - (g) Neptune - 0-3871 0-7233 I -0000 1-5237 5-2028 9-5389 30-0370 87-969 224-700 365-256 686-980 4332-585 IO759-22O 6OI2O-72O Express the periods in sidereal years true to 4 significant figures, and show that in these units the squares of the periods are the same as the cubes of the distances. (A sidereal year is 365-256 days.) 9. From the following table express the duration of the longest days in terms of the duration of the corresponding shortest day. State the answer true to 3 significant figures. Latitude. Longest Day. Shortest Day. hrs. min. hrs. min. (*) O 12 12 w IO 12 35 II 25 w 20 13 13 10 47 Id) 30 13 56 10 4 (*) 40 14 51 9 9 (/) 50 16 9 7 5 1 () 60 18 30 5 30 REVISION PAPERS 149 EXERCISES XLII. Miscellaneous. 1. Write in figures two and seven tenths, 70 and nine hundredths, 50 and forty-six thousandths, and 123 millionths. Find the sum of the four quantities and subtract it from 200. 2. Four different bottles contain respectively 27, 1-65, 5-13 and 0-5 litres of wine. How many litres are there altogether ? 3. Nine clocks of equal value are sold for 7. 17 's. 6d. What is the cost of one ? 4. How many grams are there in 2 kilograms 50 grams ? 5. How many gallons are there in 71 pints and how many quarts and pints over ? 6. From a caddy containing 7-25 oz. of tea 1-5 oz. are taken. How many ounces are left in the caddy ? 7. If the weight of a packet of cartridges is I Ib. 7 oz., what is the weight of 7 packets ? 8. A room is 10 ft. high, 18 ft. 6 in. long and 12 ft. wide ; how many cubic feet of air will it contain ? 9. Find the area (in square inches) of a sheet of paper 9-2 inches long and 6-5 inches wide. 10. How many copies of a book costing ,0-25 can be bought with 11. Add together 0-375 of 13^. 4^. and 0-07 of 2. ios., and subtract the result from 0-45 of i. 12. Find the rent of 42-75 acres of land at ^1-255 per acre. Give the answer to the nearest penny. 13. Multiply 52-08631 by 0-38056. Give the product in full, and then give it true to the nearest thousandth. 14. How many times is 0-006552 contained in 155806-56? Give your answer in words. 15. Find the cost of carpeting a room 20 ft. 3 in. long and 13 ft. 4 in. wide, with a carpet 2 feet 3 inches wide at 5^. a yard. A MODERN ARITHMETIC 16. If 3-2 pounds of sugar are taken from a bag containing 7 pounds, how much is left in the bag ? 17. Ten trucks of coal have a gross weight of 168 tons 15 cwt., and the trucks themselves weigh 39 tons 18 cwt. 2 qrs. How many sacks of coal, each containing i cwt. i qr., can be filled from the trucks, and what amount of coal remains ? 18. From the given tables find the average (a) rent and (b) population per acre in the following Welsh counties rent to nearest \d., population to four significant figures : (N.B. Take the average rent in this case as the total rent divided by the total extent in acres, and similarly for the population.) COUNTIES. No. of Acres. Population. Gross Rental. Anglesey 120,417 34,808 161,356 Brecon - 473,087 53,951 365,137 Cardigan - 595,285 82,707 414,856 Carmarthen 464,587 123,570 698,284 Carnarvon 322,742 137,236 809,253 Denbigh - 378,309 126,458 773,656 Flint 106,878 60,536 377,H3 Glamorgan 576,537 866,250 7,190,934 Merioneth 523,708 64,248 35!,205 Montgomery - 591,973 63,994 514,202 Pembroke 357,H8 82,424 457,830 Radnor - 238,660 20,241 H3,5 61 19. Add together 0-023 of a , 0-946 of a shilling and 3-48 pence, and subtract the sum from 0-26 of a guinea. Give the answer in pence and the decimal of a penny. 20. (a) Find the value of 97-96 tons of copper at 73-75 per ton. Also (&} find in Ibs. and decimals of a Ib. the weight of i bushel of corn when 32 bushels weigh 11 cwt. 2 qr. 16 Ib. 21. The edges of a carpet in a room 25 ft. 6 in. long and 18 ft. 4 in. broad are i ft. 3 in. from the walls. Find the area of the carpet, in sq. yards, feet and inches. REVISION PAPERS 22. How many times can a jug holding 0-078125 gal. be filled from a vessel containing 786-375 gal. ? What part of a pint remains over ? 23. A cistern is 6 ft. long and 5 ft. broad, and it is of such a depth that it would take 1440 bricks, each 9 inches long, 3 inches deep and 4 inches broad, to fill it : how many gallons would the cistern hold, if a pint of water contains 72 cubic inches ? 24. Complete the following table by evaluating the average return for each county of the returns under each of the given heads. Most Criminal Counties. Summary Jurisdiction. Ignorance Height of A/inif COUNTIES. Committals per TOGO. Lawlessness per looo. Drunkenness per looo. per cent, signing with marks. QUlt persons. Inches. () (V (*) (rf) w Hereford - 0-97 21-78 5-00 14-9 66-45 Middlesex - o-93 20-72 6-76 10-0 66-69 Lancaster - 0-86 27-03 13-29 20-7 67-49 Warwick 0-68 21-72 4-88 19-7 67-12 Worcester - 0-65 17-89 5-26 18-8 67-22 Bedford - 0-63 11-32 2-57 25-8 67-07 Monmouth 0-61 21-70 6-60 30-1 66-45 Cumberland o-53 1 6-08 9-33 16-3 68-48 Gloucester o-53 1 8-00 3-97 14-0 66-31 Norfolk - o-53 11-03 2-37 17-9 68-34 Northampton Stafford 0-56 0-56 1 1 -80 25-35 2-90 6-47 14-5 28-0 67-26 67-82 Average = Least Criminal Counties. 00 (*) (*)' (0 (/) Cornwall - O-2I 6-37 1-63 19-7 67-94 Huntington O-26 I3-I9 1-72 16-3 66-78 Durham 0-31 > 25-10 3-50 22-5 67-70 Cambridge 0-31 n-53 1-47 17-3 66-78 Devon 0-27 11-65 3-53 10-6 67-28 Derby 0-32 19-23 6-26 25-8 67-80 Nottingham o-33 22-42 5-50 18-0 67-38 Rutland o-33 11-85 2-19 9-4 67-29 Suffolk o-35 2-91 o-43 18-9 67-95 Westmoreland o-33 15-97 5-34 8-0 68-00 Northumberland o-34 22-70 16-20 14-5 68-59 Essex 0-36 6-10 1-61 13-2 68-00 Average = 152 A MODERN ARITHMETIC 25. If the water in a tank, 8 ft. long 7 ft. wide, is 4 ft. deep, and a cubic foot of water weigh icoo oz. : what is the weight in tons of the water in the cistern ? 26. Multiply 0-02475 by 0-64, and divide the product by 0-000125. 27. Multiply ^15. 8.?. ^d. by 7-8498, giving the answer in pounds, etc , to the nearest penny. 28. Add together 0-0021 of a cwt., 0-045 of a quarter, 0-37 of a lb., and subtract the sum from 35-263 ounces. Give the answer in ounces and the decimal of an ounce. 29. Below is a table of the speeds (some only approximate) of various animals and agencies : (a) Fill in the missing data in (iii). (d) (xi). (e) Find how long the trotting horse in (iv) would take to describe a distance of 14-55 miles. (/) How many times faster the average Atlantic liner is than the average racing yacht? (Assume I metre = 3-28 ft., and state the answer to three significant figures.) Centimetres per second. Feet per second. Kilometres per hour. Miles per hour. (i) Snail - - - 0-16 O-OO5249 0-005759 0-00358 . (ii) Hunted fly - 970 3I-83 34-9 2 21-7 (iii) Carrier pigeon - 1800 (iv) Horse trotting - I3OI . .. (v) Race horse 1670 5479 60- 1 2 37-36 (vi) Man running I4-25 (vii) Racing yacht . .. . . . 27-80 (viii) Atlantic liner . .. 24-4I (ix) Rifle bullet - ... 1900 (x) Cannon ball 91440 3000 32.93 2046 (xi) Rivers 1 0-8o ... (xii) Average wind 55 18-04 19-80 I2-30 REVISION PAPERS 153 30. (a) What is the quotient when 61-061 is divided by 0-305 ? (b) How many pieces each 1-19 of an inch can be cut off from a rod which is a yard long, and what is the length of the piece remaining ? 31. Find the Barometric Pressure at York on Oct. 5, from the given data : knowing that the average pressure for the 7 days is 29-88. Barometric Pressure at York, Oct. 1-7 inclusive. Oct. i. Oct. 2. Oct. 3. Oct. 4. Oct. 5. Oct. 6. Oct. 7. 29-45 29-60 29-78 29-79 ... 30-19 30-20 inches of mercury. 32. In a room 16 ft. 6 in. long by 12 ft. 4 in. broad, a carpet is laid down 10 ft. 8 in. long by 8 ft. 3 in. broad : how many sq. feet of floor are left uncovered ? 33. Find, in cubic feet, the content of a room 16 feet long, 10 ft. 6 in. broad and 8 ft. 9 in. high. 34. How many lengths of 0-0375 ft. are contained in 31-7296875 ft. ? Express the remainder as the decimal of a foot. 35. Find the dividend on ,3275. los. at 145-. $\d. in the . 36. Divide 3-64353 by 0-0671. Explain the working. 37. Find the average monthly rainfall at the two Irish stations below, from the data given : (a) Gal way (b) Cork (a) Gal way (b) Cork Jan. Feb. Mar. Apr. May. June. 5-92 2-39 1-62 i-53 3-02 3-H inches. 6-09 0-78 O'Of i-34 3-15 4-87 }> July. Aug. Sept. Oct. Nov. Dec. 0-95 2-60 8-30 5-88 4-88 2-73 inches. 2-28 2-42 \ 5 2 379 3-67 3-20 38. Find the average temperature during July 1903 at Hull and Oxford, given the average monthly temperatures as below : 1903. July. Aug. Sept. Oct. Nov. Dec. Hull 5S-o 55-3 50-4 43'3 37-9 Oxford ... 58-7 56-8 5 2-2 44. i 38-6 1904. Jan. Feb. March. April. May. June. Average for 12 months. Hull 39' i 38' I 40 -2 47-8 5'-3 55'4 48-o6 Oxford 39'4 39' i 4I-O 49'3 5 2 -7 57-2 49'2 154 A MODERN ARITHMETIC 39. In the accompanying table the rainfalls in inches, for the different months of the year 1874, are stated for different parts of England. Fill in the missing parts of the table. 1 / ^-\ (*) Middlesex. W Norfolk. to Cornwall. <<0 Gloucester. Lancashire. \j ) , Yorkshire, E. Riding. Jan. 1-34 0-89 5-62 3-93 3-53 1-05 Feb. - I-IO 0-97 3-86 2-40 1-59 i\33 March - 0-62 o-93 i-3i 2-14 3-37 1-18 April i-33 1-03 2-34 1-99 I-OO OI May - i -08 1-23 0-80 0-66 i-39 56 June 2-II 2-91 i -80 i -06 I -00 0-57 July - I-ll 1-52 i-53 1-56 1-77 48 Aug. 177 i-73 4-08 4-64 4-34 98 Sept. - 3-02 7-15 7-07 3-86 80 Oct. - 3- 6 4 1-30 6-ii 3-82 376 77 Nov. - 2-24 2-35 4-59 2-36 4-85 3-71 Dec. - 1-65 2-46 8-14 3-62 3-64 2-75 Monthly) I.7.C Average J / J 40. A ship, steaming towards a port in a fog, fires a signal gun which is answered from the port, as soon as heard, by another gun ; the report of the latter reaches the ship 25-5 seconds after the first gun was fired ; the ship fires again immediately, is answered as before, and this time the reply is heard in 24-5 seconds. If the sound travels 1 100 feet per second, at what rate must the ship be steaming? 41. A rectangular piece of paper is measured in inches and in centimetres, with the following results : Length, 8-73 inches ; 22-17 centimetres. Breadth, 5-57 inches ; 14-15 centimetres. Find the area of the paper in square inches and in square centimetres, and hence find the number of square centimetres in a square inch, true to o-oi sq. in. REVISION PAPERS 155 42. What is the average depth of a rainfall in which 7 million litres of water fall on a square kilometre ? 43. If a cubic foot = o-o28 cubic metre, and if I kilogram = 2-2 lb., find the number of ounces in i cubic foot of water. (The kilogram is the weight of a litre of water.) 44. A wheel makes 1028 revolutions in passing over 2 miles 4 fur. 9 pis. 5 yds. 6 ins. What is the circumference of the wheel ? 45. A gravel walk 6 feet wide runs round a grass-plot 60 feet long and 40 feet wide. If gravel is 35-. per cubic yard, find the cost of a layer of gravel on the path 3 inches deep. 46. The table below gives the average daily duration of daylight at Latitude 52 N. for each of the twelve months, with the exception of July. The mean or average duration for the months January to July inclusive is 12 hrs. 39 min. (a) What is the duration during July? (b) What the average duration from July to December inclusive ? DURATION OF DAYLIGHT. Date. Latitude 52 N. Date. Latitude 52 N. January - February - March April May June hrs. min. 7 50 9 5 10 50 12 54 14 51 16 23 August September October - November December hrs. min. 15 31 13 40 II 41 9 40 8 8 47. When 14 cwt. I qr. 18 lb. of sugar are taken away from a ton weight of it, what decimal of a ton remains ? (State your answer true to the sixth decimal place.) 48. Find the area of a room 19 ft. 4 in. long and 14 ft. 3 in. broad. What is the cost of a carpet for it, 2 ft. 9 in. wide, at 4.?. \\d. per yard ? I 5 6 A MODERN ARITHMETIC 49. Reduce 8 metres 7 decimetres 5 centimetres to yards, feet and inches, [i metre =1-09 36 yards nearly.] 50. An open tank, measuring on the outside 6 yd. 2 ft. in length, 3 yd. 2 ft. in width and 7 ft. in depth, is made with sides and floor of brickwork i ft. thick, and is filled with water. Find the weight of the tank and its contents, it being given that one cubic foot of water weighs 1000 oz., and that brick is one and a half times as heavy as water. 51. Divide (a) 8-468 by 0-0292 and (b) 0-8 by 3-175 ; state the answer true to 5 significant figures. What is the value of (c) 2-778125 of 6s. 8^+0-75 of 2s. 6d. -0-935 of 3-y. 4^. ? 52. A piece of carpet 72 yards long and 27 inches wide, costing ^12. I2s., is used in covering a room. Find the cost per square yard. 53. The table below shows the difference between clock time and apparent noon (as determined by observations on the sun) at different times of the year : Find, true to the nearest minute, the average clock time at apparent noon during each half (i.e. Jan. to June ; and July to Dec.) of the year. Clock Time at Apparent Noon. Clock Time at Apparent Noon. January i - February i - March i - April i May i June i hrs. min. . 12 4 12 14 12 12 12 4 ii 57 II 58 July i August i September i October i - November i December i hrs. min. 12 3 12 6 II 50 ii 44 ii 50 12 2 VULGAR FRACTIONS 157 32. VULGAR FRACTIONS. The term decimal fraction has already been used to express the portion that one quantity is of another. The same result may be expressed in another way. Thus dealing with 3 oz. as compared with i lb., we might suppose i lb. divided into 1 6 equal parts, of which 3 are taken, and say 3 oz. is three-sixteenths or -f\ of i lb. Similarly, if we speak of T 5 ? of a yard, the meaning is : that the yard is supposed to be divided up into 18 equal lengths, and of these 5 are taken to form the required portion. So by the vulgar fraction ~, we mean, that a certain quantity regarded as a unit is divided up into b equal parts, and of these a are taken to form the fraction, a is styled the Numerator, b the Denominator. EXAMPLE. What is the value of-f^ofi cwt. ? i cwt. could be divided up into 16 equal parts each of 7 Ibs. T 3 6- of a cwt. would be 3 such amounts, or 2 1 Ibs. EXAMPLE. What is the value of\\ of $. 2s. 6d. ? r . j We can divide $. 2s. 6d. into 25 equal - amounts of 2s. 6d. 13 such parts will amount to i. I2S. 6d. 5 ' if of $- 2S - 6d.=i. i2*. 6d 12 EXAMPLE Evaluate ^0/2 tons 3 cwt. 1 7 lb. Here we have to divide the 2 tons 3 cwt. 17 lb. by 27, and then to multiply the answer by 14. 158 A MODERN ARITHMETIC And it is clearly just the same if we multiply by 14 first and divide by 27 afterwards. The operation which is done first depends upon the particular example. Here 2 tons 3 cwt. 17 Ib. -1-27 =i cwt. 2 qr. n Ib. i cwt. 2 qr. ii Ib. x 14 = i ton 2 cwt. i qr. 14 Ib. .'. ^i of 2 tons 3 cwt. 17 Ib. = i ton 2 cwt. i qr. 14 Ib. EXAMPLE. Evaluate ^ of 2. $s. ^d. 2. y. 4<t. x 12=^26, EXERCISES XLIII. a. Mental or Oral. State the value of the following : 1. of is. 2. | of is. 3. f of is. 4. |of^i. 5. i of ios.6d. 6. -$ of 2. 7. of i ft. 8. f of i ft. 8 in. 9. ^ of 3* 6d. 10. f of i Ib. 11. if of i cwt. 12. } of 2 cwt. 2 stones. 13. $}ofs. 14. 5^ of i qr. 2 Ibs. 15. ^f of 3 cwt. 16. || of 3 yds. 17. & of ^2. 8j. 18. of a furlong. EXERCISES XLIII. b. Find the value of the following : 1. if of 4 cwt. 2. T 6 r of^i2. ioj. 3^. 3. j| of ^19. i7j. 6r/. 4. /s of 5 tons 9 cwt. 17 Ibs. 5. M of 7 Ibs. 12 oz. 6. H- of 93 miles 18 chains. 7. if of 132 yds. i ft. ii ins. 8. | of 9 m. 8 cm. 8 mm. 9. & of 8 kilogs. 4 Dekagr. 10. ^ of 40 ac. 2 ro. 11. | of a gross + f dozen - ^ score. 12. f|o f ^3- 4-f- - 1 guinea. 13. ;} of 43 yds. 2 ft. 3 in. +f\ of J chain. 14. } of 136 tons 8 cwt. + l\ of 344 cwt. 49 Ibs. - of i ton. VULGAR FRACTIONS 159 Graphic Illustrations of Vulgar Fractions. EXAMPLE. Draw a rectangle 8 cm. by 9 cm., regard this as a unit ; divide and shade portions to represent FIG. 21. EXERCISES XLIII. c. Graphic. \Apparattis: Squared Paper.~\ 1, Draw 5 rectangles 12 cm. by 8 cm., each of which you may regard as a unit ; divide and shade portions to represent the following fractions : (a) 1. (*) T V (c) if. (</) f|. 2. Taking a line AB as unit, draw lines to represent the following fractions : (i) I- (ii) I- (i") I- (iv) f. 160 A MODERN ARITHMETIC 33. IMPROPER FRACTIONS AND MIXED NUMBERS. If by means of a vulgar fraction it is wished to compare 'the magnitude of some quantity A with that of another quantity B which is taken as the unit, but which is smaller than A, the result must be either an improper fraction or a mixed number. It may be said that 33 oz. is f J of a lb., because a Ib. would have to be divided into 16 parts and 33 such parts taken to obtain 33 oz. ; the fraction ^|> in which 33 (the numerator) is greater than 16 (the denominator), is known as an improper fraction, to distinguish it from a proper fraction, where the numerator is less than the denominator. It might have been said, however, that 33 oz. is 2 lb. and i oz., i.e. is 2 lb. and -fy of a lb. ; the fraction f f would then be represented by 2 and T \, or usually 2^ ; this fraction, consist- ing of a whole number and a proper fraction, is known as a mixed number. EXAMPLE. Express as mixed numbers the following : -Jf, - 3 9 -. (1) ||. Divide by 12, obtain 2 and 4 over, i.e. f|=*A- ( 2 ) - 3 <r = 4F- Divide by 8, the quotient 4 gives the whole number, and the remainder 7 gives the numerator, while the denominator is unaltered. EXAMPLE. (a) To convert a mixed number^ say 6f, to an improper fraction. 6 units are the same as 7x6 = 42 sevenths. Therefore altogether, there are 42 + 5 = 47 sevenths, and the correspond- ing improper fraction is - 4 /. (b) Convert iiy 7 ^ to an improper fraction. IMPROPER FRACTIONS AND MIXED NUMBERS 161 EXAMPLE. Find 5^ of $ cwt. 2 qr. i.e. we have 5 times 3 cwt. 2 qr. = 17 cwt. 2 qr. and y of 3 cwt. 2 qr., i.e. 3 times 2 qr. = i cwt. 2 qr. .'. 5f of 3 cwt. 2 qr. = 19 cwt. EXERCISES XLIV. 1. Express as improper fractions : (a) 3ll (*) Si- (') 1 8ft- (<*) 2. Express as mixed numbers the following : () W- 0) W (') W- ('0 f- (/) w- (^) m*. (^) - i tp- (o m- o) w- 3. A spends 2^ times as much as B, who spends 5^. 5^. 'What does A spend ? 4. How much greater is 3/5 of ^4 than 4$ of 2. ioj.? 5. What must be added to 3 T 7 5 of 4 tons 10 cwt. to make 20 tons ? 6. When 7| lengths of 168 yards have been cut from f of a mile, what length remains ? 7. What must be taken from iSif cwt. to leave ;i of 3 qrs. I4lbs.? 8. Find the value of 3^ of ,9. gs. 4^+8^4 of 19^. 6d. - 13^ of ^i. 2j. yd. One Quantity as a Vulgar Fraction of Another. EXAMPLE. Suppose if is desired to express 2. 135. 6d. as a vulgar fraction of ]^^. 155-. 2. 135". . = 535". . 4. i$s. 0^.^95^. =1140^. 2. 135. 6d. could therefore be obtained by dividing 4. 155-. into 1140 equal parts (each id.) and then taking 642 of these. 2. 135-. 6d. is accordingly T \ 4 T 2 Q- of 4. i$s. T.M.A. L A MODERN ARITHMETIC A slightly different procedure is as follows : 2. i$s. 6^ = 535. 6d. = 107 sixpences. 4. i$s..od. = <)$s. =190 2. i$s. 6</. could be therefore obtained by dividing ^"4. 15^. into 190 equal parts each equal to 6d. and taking 107 of these. 2. i 3 s. 6</. = J8of .4- I5J . J-JJ and T 6 i 4 4 2 ir nere express the same portion these fractions are therefore said to be equivalent. 34. EQUIVALENT FRACTIONS. To express the part that 3 cwt. is of i ton, we can use the fraction $, for i ton could be divided up into 20 equal parts, termed cwt, of which 3 parts are taken ; but the ton may also be divided into 80 qrs. and each cwt. into 4 qrs., the part taken being 12 qrs. out of 80 forming the whole ton, and therefore represented by the fraction ~, i.e. 3 * 4 . Again, 3 cwt. is 80 20 x 4 112 x 3 Ibs., i ton is 20 x 112 Ibs., and the fraction could be also represented by -3 II2 20 x 112 and therefore the fractions 3 4X7 II2X-? , , - i- are equivalent ; 20 4 x 20 112 x 20 and so generally a fraction | is equivalent to the fraction n x a n x b' A fraction is unaltered if both numerator and denominator are multiplied by the same quantity, or if both numerator and denominator are divided by the same quantity. When the numerator and denominator are the simplest numbers possible, a fraction is said to be in its lowest terms. EQUIVALENT FRACTIONS 163 EXERCISES XLV. a. 1. (a) Write down the fraction expressing the part that 4 shillings is of a pound, in the same manner that the fraction fy was used on p. 162 to express the part that 3 cwt. is of i ton. Write down also the equivalent fractions possible on regarding a shilling as -'equivalent to (b) 2 sixpences, (c) 4 threepenny bits, (d) 12 pennies, (e) 24 halfpennies. 2. Express as equivalent fractions in their lowest terms : (*) i% (*) i2, to sV (rf) &, (*) A, (/) iVb, Or) IJ. 3. Group together the equivalent fractions among the following : &, M, ft, , &fc If, , If, 1%, &, A- 4. Write down 4 fractions equivalent to the following, (i) with smaller numerators, (ii) with larger numerators : () *f. (*) *! W Ml- 5. Fill in the missing parts (numerator or denominator) of the following fractions : so that they may be equivalent fractions to (a) . (*) 2- W I 6. What weight is the same fraction of 28 Ibs. as (a) 84 Ibs. is of 112 Ibs., (b) 15-y. is of ,4. 49., (c) one yard is of 4 ft.? 7. What length is the same fraction of 36 inches as (a) an area of 2 sq. ft. is of I sq. yard, (b) a volume of 432 cubic inches is of a cubic foot ? 8. One person has ^360, another ,108, to give away to charitable institutions. If the first gives a hospital ^80, what does the hospital receive from the second person if he gives to it the same fraction of his total gifts as the first ? 164 A MODERN ARITHMETIC 9. At one match a cricketer A made 54 runs out of a total of 297, and he made the same fraction of a total of 264 at a second match. What was the fraction, and what number of runs did he make at the second match ? We can see by means of equivalent fractions which of two or more fractions is the greatest. Thus : EXAMPLE. To find out which of the two fractions -|f, \\ is the greater. We try to find two fractions equivalent respectively to the given fractions, but with equal denominators. Thus J, ff, f \ , T V, T 6 3 5 5 are all equivalent to if Also JJ, *, yVV, ^ are all equivalent to . f^s is clearly greater than -/^V* an d therefore If- is greater than JJ. The numerators might have been made the same as in the following worked example : EXAMPLE. To find which is the greater of the two fractions T? F> idr F> dhr> r& TJTT> <mr are a11 equivalent fractions. Ts> -f& AT> T 2 A form another group of equivalent fractions. T VV is clearly less than J^TJ whence T f is less than T f --. EXERCISES XLV. b. 1. Express the fractions Q, f, ]f with the same (least) common denominator ; then arrange them in order of magnitude. 2. Write down the greater fraction of the following pairs : () H 7- (*) If 2*. (0 H, e- <<*) 4, *f. W H, M- (/) iiS, W (^) #* ffl. (A) M, -|. (0 H, if- 0') M, M- (^) T 4 f, H- EQUIVALENT FRACTIONS 165 C FIG. 22. Equivalent Fractions Illustrated Graphically. Draw any line AB (Fig. 22), and regard it as a unit. Bisect the line at C, then AC represents the fraction \. Divide AB into 4 equal portions. AC F H K contains 2 of these, and hence - AC also represents the fraction A f. Divide AB into 6 equal portions, say at E, F, C, H, K. AC contains 3 out of the 6, and therefore represents the fraction f . The fractions ^, f, f being all represented by the same length AC are hence all equivalent. Again, draw a rectangle ABDC (Fig. 23) 6 cm. long, 4 cm. broad, and regard it as a unit. AEFGHIB A F DKLMNOD C L FIG. 23. FIG. 24. FIG. 25. The rectangle, may be divided into 6 equal portions by the 5 lines EK, FL, GM, HN, IO, and AFLC represents the fraction f . In Fig. 24, AFLC represents the fraction J. In Fig. 25, AFLC represents the fraction ,f T . Hence the fractions f, J, -% are all equivalent. EXERCISES XLV. c. Practical. [Apparatus: Squared paper. Scales. Set squares] 1. Draw a line 3" in length and regard it as a unit ; mark off a portion to represent the fraction y\ ; show from your figure that the fractions f, are equivalent. 166 A MODERN ARITHMETIC 2. Take a line AB 4" in length as a unit ; cut off from it portion AX to represent the fraction of the unit, and show that it is the same value as j 6 6 . 3. Draw a square of 6 cm. side, and consider it as a unit area. Shade by diagonal lines f of the area. Shade by cross diagonal lines j% of it. Note that these areas represent equivalent fractions. 4. Draw a rectangle 6 cm. long, 4 cm. broad ; regarding it as a unit, mark by shaded lines an area representing the fraction f ; show graphically that the fractions f , T 9 ^, -f are equivalents. 5. Show graphically, by suitably divided squares or rectangles, that in the following sets the fractions in each set are equivalent : () i A, it- (*) 1% I M- to M, I, A- 35. ADDITION AND SUBTRACTION OF FRACTIONS. Two or more fractions with the same denominators can be easily added together. EXAMPLE. To evaluate ||- + -|f + If This addition means the sum of 17, 18, 32 equal parts each (^V) th f a umt > *'* there are 67 such portions. Hence the sum = n 3514_21_1 T ~ TIT T2 If the fractions have different denominators, they may be changed into equivalent fractions in which the denominators are the same. ADDITION AND SUBTRACTION OF FRACTIONS 167 EXAMPLE. To evaluate f + T V + T S F- l i?> i lf> !i> H> f are a11 equivalent fractions; so are 14 21 28 orr1 alc/~> 369 J -2T> ae'J 48' and also TF 15 2 J TF The sum f + T 7 ^ + j 3 g- is therefore the same as ' EXAMPLE. To find the difference between f f and |. 38 4_38 2 8 _ 1 _ 2 ST ~ 5" ~ 3T ~ ~5 ~5 IJT T If the fractions be mixed numbers, the whole numbers should be considered separately. EXAMPLE. Evaluate 3]- + 2\ + if- + . EXAMPLE. Subtract $l s from 6-J, z>. evaluate 6|--3-J. A3 ^7_/;i3 ^ 7_/^ ->i3 7_^ i 27 4\) OT ~ STF - b + T ~ 3 ~ - b ~ 3 + 7- ~ -y - 3 + o-;j ~ where f and ,j are changed into equivalent fractions with the same denominator. Since J- is greater than f |-, proceed either thus r -I 27 49_ i T i 27 49_^ i 90 3 + FIT ~ F 3" 2 + "*" FU' ~ F"3 2 "I" 3 or thus EXERCISES XLVI. a. 1. One weight is 3* cwt. and another is 2-J^cwt. Express these in cwt. and Ibs. ; find their sum. State the preceding answer in cv.'t. and fractions of a cwt.. and verify your result by adding together the fractions 3} and 2^f. 2. Find the sum of ^2}f and ^5^ ; verify in a manner similar to that suggested in Exercise i. 168 A MODERN ARITHMETIC 3. Add together the following : () I, 2fr, 3 |, *. <*) 2|, , I, 3 J. (0 tt 3t, 61 #. (d) 73, 8|, i|, 2|. (0 3&, 4H 41, IS- (/) 1 3t, 4t, 5 2 7 <J, * te) 2oj|, is, i&, H, '9&. (*) 3i, 4*, 93tt, 2 f> tt 4- (0 2tf, I8H, I, if, f ft. 0) 3A, 72j, 8i, 2oJ, gg, i6g. Find the difference between the following pairs of fractions : 4. (a) I and ft, (*) f and |, (f) } J and & (^) ^ and |. 5. () 3f and li, (d) ft and f , (^) ft and T 5 3 , (rf) ^ ff and If. 6. (a) .11 and f 8, (*) ^ and ||, (^) JL and _7_ (d} ^ and ^ 7. (a) i 9 /j and 13^, (^) 20}^ and 17^^, (t) i8 T % and 4 oJ, (rf) 760^ and 63^. 8. Evaluate | - + g J - 1 + 1 - f |. 9. Evaluate 10. Evaluate 11. Find the value of ^7 + 4* guineas i8f hah guineas + 14/0 crowns. 12. To the sum of 6| crowns and || of a sovereign, add the difference between 8| shillings and $ J. 13. What is the value of 13! tons+i6f cwt. -3iJ| qr. - 143 Ib. ? 14. Take I7j 5 j of a furlong from 5| miles. 15. When 23$ yds. of a rope had been used, it was noted that 19^2 ft. were left. What was the length of the rope ? EXERCISES XLVI. b. Practical. [Apparatus : Squared paper. Set squares. Scales."] 1. Draw a line AB say 2" long, divide it up into 3 equal parts ; take tiuo of these parts to represent the fraction | ; next mark off on your paper a line twice as long as AB, and divide into 3 equal parts ; compare the length of one of these parts with the line representing the fraction |. ADDITION AND SUBTRACTION OF FRACTIONS 169 2. Draw a line AB 5" long ; divide it into 4 equal parts ; let the points of division be A 1} A 2 , A 3 . Next divide the portion AA 3 into 3 equal parts, and give the value of of this. Note that it coincides with | of original, and hence of = ...? 3. Draw a line 5" long. Divide it into 4 equal parts by repeated bisecting ; mark off ^ of the line, call it XY. Divide this portion XY into 5 equal parts, and mark off a part containing 2 of these ; letter it XZ. Then XZ is f of 2- Divide the original line into 10 equal parts and show that XZ is 3 of these or j 3 ^ of original line. Hence ? of 2 = 'iV 4. Draw a square of 9 cm. side and letter it ABCD. Divide AD into 3 equal parts at F, G. Through G draw GE parallel to AB. Then ABEG is of the square. Divide AB into 9 equal parts at H, K, L, M, N, P, Q, R. Through N draw NO parallel to AD. Then ANOG is | of ABEG, and ABEG is of the square. If the number of square centimetres in ANOG be reckoned, they will number 30, and .'. ANOG is |^ of the whole ABCD. So of f = J$ or f 5. Draw a rectangle 2" by i" ; divide it up into two unit squares (each square of I sq. in. being regarded as a unit). Divide up the rectangle also into 3 equal portions by 2 lines parallel to the 2" side ; show from your figure that J of 2 is . 6. Draw figures to show that is equivalent to \ of 3 ; that 4 is equivalent to \ of 4 ; and that of 3 is the same as f . 36. CONVERSION OF FRACTIONS. Conversion of a Decimal into a Vulgar Fraction. EXAMPLE. To convert 0-216 to a vulgar fraction. 0-2 we know when multiplied by 10 = 2, and is therefore the tenth part of 2, or ^ or ^^ or 1%^, these being equivalent fractions. 170 A MODERN ARITHMETIC o-o i becomes i when multiplied by 100, i.e. it is the hun- dredth part of unity or T J<y. 0-216 when multiplied by 1000 becomes 216, and is therefore the thousandth part of 216, which is the same as = -.-. 1000 io 3 Similarly, (a) 32-3452 = ; From this process it is seen that, when decimals are expressed as mixed fractions, (i) whole numbers remain whole numbers ; (ii) the numerator of the fractional part consists of the decimal figures ; (iii) the denominator contains 10 raised to a power equal to the number of decimal figures. When the decimal is to be expressed as an improper fraction, the complete decimal is written down without the decimal point to form the numerator, and the denominator consists of TO raised to a power equal to the number of figures after the decimal point. Thus : 518-237 = 51^37 = S1?37. ' I0 3 1000 Conversion of a Vulgar Fraction into a Decimal. EXAMPLE. To convert into a decimal fraction. The fraction means that a certain quantity regarded as a unit is divided up into 60 equal parts, and of these 24 are taken. But we have seen that the same result is obtained by dividing a quantity consisting of 24 units into 60 equal parts. Hence, to convert J into a decimal, we merely divide 24 by 60 by the ordinary methods used in decimals ; and similarly, in converting any vulgar fraction - into a decimal, b we merely divide a by b. CONVERSION OF FRACTIONS 171 Recurring Decimals. EXAMPLE. Convert f |- into a decimal. -f-J- when reduced to lowest terms = T V Dividing 8 by 15, we obtain 0*533 T ne remainder is continually 5 and the process will never terminate ; the decimal quotient is re- presented by 0-53, where the dot above the 3 5 indicates that the 3 can be repeated indefinitely. Similarly, 21-87643 would represent a non-terminating or recurring decimal, in which the figures 7643 recur indefinitely. 21-876437643764376437643 .... EXERCISES XL VII. a. N. B. In no case need the decimal equivalents of any of the fractions stated be carried to more than five figures. 1. Express the following decimals as vulgar fractions, reducing the latter always to lowest terms : (a) 0-6. (b) 0-06. (c} 0-006. (d) 0-125. (*) 0-75- (/) 0-32. Cr) Q-375- (/O o-55- (0 0-0675- (j) 0-664. (k) 0-875. (<0 0-0015625. (;) 7-64. (n) 590-875. (0)0-06325. (P) 078575- (?) 0-0525. (r) 0-9816. (s) 41-1875. (/) 0-09375. () 0-7925. (?/) 0-003875. (w) 18-6875. (x) 62-01025. 2. Arrange the following vulgar fractions in descending order of magnitude by first converting them into decimals ; check your results by the method of equivalent fractions : (*) &, I ft* & (*) II ^ ^ 5- 172 A MODERN ARITHMETIC EXERCISES XLVII. b. Graphic. [Apparatus: Squared paper. Diagonal and other Scales I\ 1. Draw straight lines to represent the following lengths : }'> V 1 ") "" 2 f"; measure the distances off also (to two decimal places) using your diagonal scale ; check your results by converting the given fractions into decimals. 2. Draw four parallel horizontal lines having their left ends vertically over one another, and measure off from the left the following lengths : 3", 3", 3-//, 3rW- Arrange the lengths in descending order of magnitude and write opposite to each its length expressed as a decimal. 3. Perform the following operations graphically : (a) if-2i + 2i + 3 |. Check your result by evaluating the sum as a vulgar fraction, and also by expressing each fraction as a decimal and calculating as a sum in decimals. Also ( 37. MULTIPLICATION AND DIVISION OF VULGAR FRACTIONS. Multiplication by a Whole Number. EXAMPLE. To multiply J by 5. f represents 3 portions, each portion being one-fourth of a unit. x 5 represents therefore 15 portions, each portion being one- fourth of a unit, and the result is therefore represented by - 1 /-, i.e. by -^p-; similarly, any vulgar fraction ~ x any whole o VULGAR FRACTIONS 173 number n = , , i.e. to multiply a fraction by an integer .we may multiply the numerator by that integer and keep the denominator unchanged. EXAMPLE. To multiply \\ by 5. As before, the result is -}- ; becoming, however, y- on reducing it to its lowest terms ; i.e. we have divided the denominator by 5. EXAMPLE. To multiply 2 T \ by 7. This is the same as multiplying T \ by 7 and adding the result to 7 times 2 ; The operations usually are performed mentally if the numbers are small. Thus 7 times 3 = 21 ; n into 21 yields 1 and 10 remainder. 7 ,, 2 = 14, and this with the 1 yields 15. And the answer is Division by a Whole Number. EXAMPLE. To divide f by 3. Here it is required to divide into 3 equal parts an amount made up of 5 equal portions, each portion one-eighth of the complete unit. If each one of these 5 por- tions be divided into 3 and one part taken from each portion, there will now be 5 parts each one twentyfourth of the unit, - - i.e. the result of the division _s_ FIG. 26. To illustrate the division of b u 4 * . a fraction. This operation can be illus- trated by a diagram in which ABCD represents a certain unit, while AEFD represents f of that unit. AGHE is \ of AEFD, and is clearly ^ of the whole. It 174 A MODERN ARITHMETIC EXAMPLE. To divide T 6 T by 3. In this case it is clear that when we divide by 3 the portion consisting of 6 parts (n of which parts make the complete unit) the result will be 2 of these parts, i.e. T 6 T -h 3 = T 2 j. Hence to divide any vulgar fraction ^ by an integer <-, either b ' (i) multiply the denominator by c, (ii) or divide the numerator by t, a a a-r-c i.e. j + c=-, or = r-> b b x c b In the case of mixed numbers, it is frequently most convenient to reduce to improper fractions before division. EXAMPLE. To divide i T 7 ^ by 3. T 7 .^_19. _19 I TS ~ 3 T2^ ~ 3 3 6"' EXERCISES XLVIII. a. Practical. [Apparatus : Squared paper. Scales. Set squares^ 1. Draw a line ^ of an inch in length ; mark it off 5 times on another line about 3" long ; show by using a suitable scale that a s>x 5 is 2|i. 2. Find graphically the results of the following operations : (a) 3 fcx2. (b) ix 3 . ( C ) 23x3. (rf) i$x3. (') J*7. (/)f xi a (g) |x 4 . Wfx7. (0^x 7 . (/) if|x 3 . 3. Draw a line i|" long ; divide it into (a) 5, (^) 3, equal parts, and find graphically (a) 1^-7-5, (b) if -7-3. 4. Draw a line 3" long; and find graphically (a) 3-r3, (*) 3-2-9. 5. Draw a line 5f" long ; divide it into 4 equal parts, finding thus, 5f-r4. VULGAR FRACTIONS 175 6. Draw a rectangle 8 cm. long and 4 cm. broad ; consider it as a unit ; mark off an area representing of the unit ; find graphically the result of dividing the fraction by 5. 7. Use the method suggested by Ex. 6, to perform the following operations : EXERCISES XLVIII. b. 1. Find the value of || x 5, and to verify your answer consider the unit as ^i, find the value 01 %\, multiply the result by 5 and express the answer as a fraction of a . 2. Evaluate (a) MX 7. W 0*5x5. (c) Vxi4- To prove your results replace the vulgar fractions by their equivalent decimals ; then multiply and change the answer back again to vulgar fractions. 3. Evaluate () IK?- (*) it-5- (0 li-6. (rf) fg-5-9. (*) if -5- 4. Prove your result by a method similar to that suggested in Ex. 2. Multiplication of one Fraction by another. EXAMPLE. To multiply one vulgar fraction - by another -. u a We have to divide ( -) into d equal portions ; each of which must be - - -, and then take c such portions. The answer . l>xd a a*c therefore = c x -. -- , = .. b x d b x d Or, the numerators are multiplied together to form the new numerator, and the denominators are multiplied together to 176 A MODERN .ARITHMETIC form the denominator; the resulting fraction is generally expressed in its lowest terms. Thus : EXAMPLE. iff x J 128 X 8l 2X1 2 Dividing numerator and denominator by 64, and also by 81. EXAMPLE.-^! x|A 175x99 In simplifying it is well to note the common factors. 5 clearly divides 35 and 175; n and 3 both clearly divide 264 and 99. Thus the fraction - ** x * x 8xjrx7 _ __ ~ gx 7X5x^x^^x3 ~~ 15' The operation of simplifying by cancelling is frequently conducted as follows : This is a clumsy process, and is productive of errors if carried at all far. EXERCISES XLIX. a. Practical. [Apparatus: Squared paper. Scales J] 1. Rule your paper into squares of -J inch side, and find the areas of the following rectangles : (a) Length ", breadth f". (/;) Length -J", breadth |j". or) i", r- (A) i", r. 2. Rule paper into squares of ^ inch side, and find the areas of the following rectangles : (a) ^' long and T V broad ; (b) f" long and -fa" broad. Find also the area of a square, (a) fy" side ; VULGAR FRACTIONS 177 3. From a square of 10 cm. side, a rectangle 9 cm. long by 8 cm. has been taken away. What area remains, and what fraction of the whole is it ? EXERCISES XLIX. b. 1. Multiply (a) f by Jf, (b) | by ft, (c) jf by JJ, (d) ft by ff . 2. () ft by 33, (*) M by i6|, (*) ^ by 27}, (O by ff. 3. (a) | of 4fr by ft of $, (J) 5fV by / n of ft, (<:) 3^ by 4. (a) || of ft of f by 2| of 4 |, (^) ^ of i|| by ^j|f. 5. Find the continued product of (a) 7?, i6f, i^ ; (b) f, H; 6. Find the continued product of ^ff, ff , ^, JJJ. EXERCISES XLIX. c. Practical. {Apparatus required: Mathematical instruments. Scales. Squared paper, ,] 1. Using your scale, find how many times \ of an inch is contained in 3 inches. Hence, write down the result of the operation 3-1- 2. Similarly, obtain U") 3KiV W 3t-T 6 o- (0 4KfV (/) 3^A- 3. Draw two rectangles, each 7 cm. long and 6 cm. broad j regarding each of these as a unit, rule them off so as to represent the fractions f and ; and from your figures compare these fractions, i.e. find what multiple f is of , or find f-r^. J.M.A. M !;8 A MODERN ARITHMETIC 4. Draw two rectangles, each 15 cm. long and 8 cm. broad ; regarding each of these as a unit, rule them off so as to represent the fractions T ^ and f ; and from your figures compare these two fractions, i.e find what multiple ^s }S f t- 5. Similarly, find by drawing suitable figures the multiples that (a) f| is off, and also of f ; () \} is off, and also of . 6. Find graphically, (a) t-s-$ ; (6) &*f . EXERCISES XLIX. <X 1. How many Ib. are there in (a) f of a cwt, (b) f of a cwt. ? What multiple is (c) f of a cwt., of f of a cwt. ? 2. What multiple is (f of a yard) of ( of a yard) ? 3. What multiple is (f of a yard) of (f- of a yard; ? 4. How many times is f of a contained in f of a ? 5. How often is $ contained in f of a , ? 6. How many times must -% be repeated to make f of a guinea? 7. How often may 2f qr. be taken from 4| ton ? 8. How many lengths of 4^ s of a yard can be cut from 38^ yd.? 9. What part of 3.1 miles is 220 yd. ? 10. How often is ft. contained in 5 yd. ? 11. How often is f pint contained in 3! gal. ? 12. How often is f of a dozen contained in 3^ scores. Division of One Fraction by Another. EXAMPLE. To divide -f- by f. Put differently the question is : What multiple is -f- of f ? . 2i| of m . 40 of 7x8 8x7 56 56 VULGAR FRACTIONS 179 Now, it is clear that if -f | be divided up into 2 1 equal parts, each - v , and if 40 of such parts be taken, the result will Accordingly, |f is |f times f , i e 5 . 3 = 5 x 8 . 3 X 7 = 5*8 = 40 "7*8 7x8*8x7 3x7 21 EXAMPLE. IJ-*-?- = i3 x 7 . 9 X 2 Q 20 x 7 ' 7 x 20' _ 13 x 7 9 x 20 140 140 And (13x7) one-hundred-and-fortieths is ^ Z times (9 x 20) one-hundred-and-fortieths. 9 x 20 _ _ 20 ' 7 20 x 9 1 80' Similarly, any vulgar fraction ~ may be divided by another vulgar fraction -. Thus d a c ay. d b x c a x d whence is obtained the following rule : To divide one quantity by a fraction, invert the divisor and multiply. EXAMPLE. i|| ^4l _i56 15 _ 12 x 13 x i5_ 12 _4 ~225 X i3~i5 x i5 XI 3~i5~5' If one of the fractions be mixed, it should be first converted into an improper fraction. o A MODERN ARITHMETIC EXERCISES L. 1. Divide ^& by (a) ft, (b) &, (c) ft, (d) 3 |, (e) AV 2. Evaluate : (a) 2fH-f. (*) MKIt- (0 iY?-?- (<0 MK'H- (') 'iVs-^Sf- 3. Evaluate, checking your results by decimals when convenient (a) 13-5-4. (*) 3f-iV to 4&-M1*. 4. Evaluate : 00 IH(i O EXERCISES LI. Miscellaneous. 1. Express los. 6d. as a fraction of 15^. State the fraction in its lowest terms. 2. Express 2. 2s. 6d. as a fraction of j. is. %>d. 3. Express three and a half guineas as a fraction of ^5. $>s. 6d. 4. When 45. jd. has been what fraction of the bill remains ? 5. Express 7 ton 17 cwt. 2 qr. as a fraction of 18 tons. 6. Express 20 yds. 28 in. as a fraction of i chain. 7. On an estate plan a rectangular block I2^"x 8j" represents 56 sq. yds. 24 sq. in. ; what fraction is the plan, of the area it represents ? 8. Express 2 qts. i pint as a fraction of 36 quarts. VULGAR FRACTIONS 181 9. A person travelled 56 miles by rail, 24 miles by coach and completed the journey of 112 miles by riding. Express as fractions of the entire length of the journey, the distances covered by each method of travelling. 10. When a cargo of 252 tons had been loaded in a ship, it was found that ^ s of the available tonnage was used ; what was the total tonnage ? 11. What fraction is 8 Ib. 3 oz. of 2 Ib. 14 oz.? Express the answer both as a mixed number and as an improper fraction. 12. What fraction of 2. Ss. is 1. 6s. &/.? 13. How often is }\ contained in the sum of f and f ? 14. Divide the sum of ^ and -- by the product of 3| and f. 15. What is the difference between the product of \% and 7!, and the sum of 2f and 3f ? 16. What must be added to the product of |f and |f to make f ? 17. Arrange the following in ascending order of magnitude : 18. By what quantity must the sum of J, f, be multiplied to yield if!? Evaluate the following expressions : 182 A MODERN ARITHMETIC 23. Find the average Bank of England minimum rate of discount for each of the years from 1893 to 1903. State your answers as mixed numbers. BANK OF ENGLAND MINIMUM RATE OF DISCOUNT, 1893 TO 1903. MONTHS. 1893- i8 94 . 1895. 1896. i8 97 . 1898. i8 99 . 1900. 1901. 1902. 1903. January - 2 r>0 3 2 2 3 3 3f 4U 4H 3o 4 February - ^ ^ 2 2 3?r 3 3 4 JS 3rV 4 March 2 2 2 2 3 3 3 4 4 3 4 April 2\ 2 2 2 2 8 3tf 3 4 4 3 4 May- & 2 2 2 2 4 3* 3 3tt 4 3 3-t June 3 2 2 2 2 3 3 3$ ft 3 3i July- - August - 2| 4 2 2 2 2 2 2 2 2 2 2 2 2" f 3f 4 3 3 3 3 3 3 September 4l 2 ^> 2 T 7 5 2* 2f 3i 4 3 3 3& October - 3 2 2 H 2J 3^ 4^ 4 3 3W 4 November 3 2 2 4 3 4 5 4 4 4 4 December 3 2 2 4 3 4 6 4 4 4 4 Average 24. Evaluate (a) |fK fit f 25. Evaluate ( fl ) r ^ 3r - T ^ 1 . 26. Reduce to their lowest terms : () If-If- (*) 38. GREATEST COMMON MEASURE. When a fraction has to be reduced to its lowest terms, the numerator and denominator are divided by all the factors which may be common to both. In most of the fractions dealt with so far, the process has been simple. We will now consider cases where the factors are not quite so obvious. GREATEST COMMON MEASURE 183 If both numerator and denominator can be split completely into their " prime " factors, it will be easy to recognize those which are common. Prime Factors are those which are incapable of being split up any further. We must begin, therefore, by stating rules which determine whether or not, some of the more common numbers are factors. (a) 2. All even numbers are divisible by 2. (b) 4. If the number expressed by the last two digits is divisible by 4, the number itself is divisible by 4. (c) 8. If the number expressed by the last three digits is divisible by 8, the number itself is divisible by 8. (d} 3 or 9. If the sum of the digits be a multiple of 3, the number itself is a multiple of 3, and this rule applies also to 9. Any number is a multiple of 9 if the sum of the digits is a multiple of 9. (<?) 11. Find the difference between the sum of the digits in the units, hundreds, tens of thousands, and other odd number of places, and the sum of those in the tens, thousands, and other even places. If this difference be o or a multiple of n, then the number is divisible by u. (/) 5. Every number ending in 5 or o is a multiple of 5. (g] 25. Every number ending in oo, 25, 50 or 75 is a multiple of 25. Proof of (<?). Consider any number such as 31742. It may be regarded as 3 x 10000+ i x 1000+ 7 x 100 + 4 x 10 + 2. Thus =3(9999 + i)+ 1(1001 - i) + 7(99+ i) + 4(u - 1)4-2 = 3(11 x 909)4-1(11 x 91) + 7 (11x9) + 4(11) + 3-1 + 7-4 + 2. = 1 1 (3 X 909 +1X91+7X9 + 4) + 3-1 + 7-4+ 2. On dividing by 1 1, the remainder is the same as that obtained on dividing 3- 1 + 7-4 + 2, and the number can be divided exactly if 3-1 + 7-4 + 2 can be divided exactly by 1 1, and similarly for any other number. 1 84 A MODERN ARITHMETIC EXAMPLE. To simplify yf ff J, i-t- to reduce the fraction to its lowest terms. 10296 = 8 x 1287 = 2 3 x 1287 8 divides the number ex- pressed by the last 3 digits, i.e. 296. = 2 3 x 3 2 x 143 The sum of the digits in 1287 is clearly a multiple of 9. = 2 3 x 3 2 x 11 x 13. 143 is seen to be a mul- tiple of i T Y i-4 + 3 = o- The expression is now split up into its prime factors. 12420 = 4 x 3105 = 2 2 x 3105 4 divides the number ex- pressed by the last 2 digits, i.e. 20. = 2 2 x 5 x 62 1 3105 ending in 5 is divisible by 5- = 2 2 x 5 x 3 2 x 69 621 has the sum of its digits equal to 9. = 2 2 x 5 x 3 3 x 23. 12420 is now split up into its prime factors. . io296_ 2 3 x 3 2 x ii x 13 2x11x13 286 12420" 2 2 x5x 3 3 x23 ~ 3x5x23 "345" EXERCISES LII. a. 1. Split up into prime factors the following numbers : (a) 225. (b) 132. (f) 792. (*9 1375- (') 504. (/) 1024. (g) 256. (A) 384- (0 320. 2. Simplify the following fractions : , . 2 3 x 3 x 5 ,,,9x11x7 . 3960 3x2x5 3 3x 8xii 5445 . ,, I22T 960 , . 524 ' 2332' (e) 528' ^ 1779' (jr) 936 , 999 /^ _99 715' 2035' v ; 999' GREATEST COMMON MEASURE 185 Simplify the following fractions : 570 . 9999 9999 912 / \ ! 2 , W 567" < w - 7 . ' 5049 DEFINITION. The greatest number which can divide exactly any given numbers is called their Greatest Common Measure. EXAMPLE. To find the Greatest Common Measure of 15000 and 4400. Resolve the numbers into their prime factors, i.e. 2 3 x 3 x 5 4 and 2 4 x 5 2 x n. Clearly the common factors are 2 3 and 5 2 , and the Greatest Common Measure = 2 3 x 5 2 = 200. EXERCISES LII. b. 1. Find the Greatest Common Measure (G.C.M.) of the following : (a) 2 3 x 3 2 x 5 and 2 2 x 3 3 x i r. () 2 4 x 3 3 x 5 2 and s 4 x 3 3 x 2 2 . (c) m and 370. (d] 1728 and 1584. (e) 1325 and 3125. (/) 252 and 297. (g) 384, 480 and 576. 2. The frequencies of the vibrations of the notes comprising the major chord C, E, G are : C, 256. E, 320. G, 384. What are the three simplest numbers which can be employed in finding the ratio of these frequencies ? 3. Draw up a table of all the prime numbers between I and 100. 4. A number lying between 1000 and 10000 is set up in type, the type becomes mixed, one figure is lost, the others being 3, 6 and 4 ; the number is, however, known to be divisible by 72. What was the missing figure, and what the actual number? 1 86 A MODERN ARITHMETIC When in endeavouring to find the G.C.M. of two numbers, the factors are not easily found, a more general method has to be adopted. Its principle will be seen from the following graphical examples : Graphical Introduction to General Method. Suppose ABCD (Fig. 27) represents the plan of a room 84 feet long, 57 feet wide, and that we wish to cover the floor of the room with large square tiles none of which are to be cut. What is the biggest tile which can be employed ? 10 20 40 50. E 60 70 80 B Mark off a distance equal to the breadth AD along the length AB, reaching as far as E say. Then, since the tiles fit exactly as far as D, they will extend exactly as far as E, and the edges of some of the tiles will therefore form a line down from E parallel to AD; draw this line; let it be EF. Then EBCF can be floored exactly with the same sort of tiles as the floor ABCD. GREATEST COMMON MEASURE 187 Set off distances, equal to EB, along BC once, twice reaching as far as G; through G draw GH parallel to CF, meeting EF at H. Then the tiles must cover GCFH exactly. Mark next FH along FC ; this can be done once twice... 9 times exactly, and GCFH can be floored exactly with 9 tiles, each 3 ft. in length ; and 3 ft. is the side of the greatest tile which can be employed to cover the floor and 3 is the greatest common measure of 57 and 84. EXERCISES LII. c. Graphic. [Apparatus : Squared paper. ~\ 1. (a) A room is 11-7 feet long and 9-1 feet broad. If its floor is to be covered by square parquets, find the greatest size of a parquet in order that no parquet need be cut. Show on your drawing (to any convenient scale) the plan of the floor with its parquetry. Find also the largest parquet which could be used for a room, (b} 13-6 ft. long 11-9 ft. broad, (c) 15-4 ft. long, 12-6 ft broad. 2. A room is 16-5 feet long by 10-5 feet broad. Find the length of the largest pattern that may be employed on a frieze, in order that the pattern may repeat an exact number of times on the frieze. 3. The circumference of one wheel is 114 millimetres, the circumference of a second is 84 millimetres. If each wheel has to be cut into teeth of the same size, what is the greatest width of a tooth ; state also how many teeth there will be on each wheel ? 4. Find the G.C.M. of (a) 87 and 63. (&) 64 and 76. (c) 135 and 215. 5. A rectangle is 15-5 cm. long and 11-5 cm. broad ; find the least number of exact squares into which it might be cut. 1 88 A MODERN ARITHMETIC General Method. The general method for finding arithmetically the G.C.M. of two numbers is now clear. EXAMPLE. To find the G.C.M. of 1794 and 4069. 1794)4069(2 Divide 4069 by 1794; the quotient 481 is 2 and remainder 481. The required g v "/ G.C.M. is also the G.C.M. of 481 and 1794. W. Divide 481 into 1794; the quotient is 3, remainder 351. The G.C.M. is there- 351 ) 481 ( i fore also the G.C.M. of 351 and 481. 130 Proceeding in this way, we see, finally, \~I7Y / 2 that the required G.C.M. is the Greatest Common Measure of 13 and 39. 91) 130(1 39 But 13 divides 39 exactly ', and hence . ^1 13 is the required Greatest Common J 3 ) 39 ( 3 Measure. NOTE. The quotients need not be written down, and the whole work may be shortened into the following form : 1794)4069 91 130 ^3 39 The work may frequently be further shortened. When at any stage factors which are clearly not common can be recognized, these may be disregarded and discarded. Thus, in the preceding example, at the stage 91 130 the irrelevant factor 10 can be discarded, and 13 is at once seen to be the G.C.M. GREATEST COMMON MEASURE 189 EXERCISES LII. d. 1. A boy has two sticks, one 144 inches long, the other 78 inches long ; he wants to cut them up into an exact number of pieces, each of the same length. What is the length of the longest piece that he can obtain in this way ? How many such pieces will he then have ? 2. Two persons A and B are distributing sums of money to poor people, each person receiving the same amount. If A distributed 21. i$s. and B distributed ,8. 5-r., what is the highest possible amount that a poor person could have received ? 3. Find the G.C.M. of (a) 1965 and 2227. (b} 1703 and 3799. (c) 1631 and 2796. (d) 1635 and l8 53- (*) 7 62 5 and 8i75- 4. Reduce to their lowest terms the following fractions : (a) 1523. (/>) H54. 2595 2912 3354 3542 XV 1024 x,v 951 5 9 04 ' 4 l8l " 21 12 ' 6023 28O5 x -\ 772 XT\ 2O4 ,ft 336 43oT* 965' "595* ^oi' x x 36l2 , v 2117 x .v 2923 y " J 2117' 10668' 3431' 5587' 5. There are three glasses of volumes 336 c.c., 90 c.c., 528 c.c., respectively. Find the greatest volume of a measure (known but not graduated) which could be employed to determine their volumes if these volumes had not been known. 6. What is the smallest number of equal weights which can be obtained by dividing up completely, a weight of 1572 Ibs., a second of 1392 Ibs., a third of 1 140 Ibs. and a fourth of 1044 Ibs. ? 7. A piece of timber 375-0 inches long, 34-5 inches broad and 24-0 inches thick is to be cut completely up into a number of equal cubes. Find the size of the greatest possible cube. igo A MODERN ARITHMETIC 8. What is the least number of equal cubes which can be arranged so as to form a block 473 inches long, 176 inches broad, 143 inches thick ? 9. A portion of a breakwater is to be 121-6 ft. long, 53-2 ft. broad, 22-8 ft. thick, and is to be built of the largest cubical blocks of granite possible. Find the size of the blocks to be used. 10. There are three heaps of bricks numbering respectively 6194, 2282 and 7498, and they are to be carted away by taking away each time the same number of loads from each heap. What is the greatest number of bricks that can be taken per load, and what will be the total number of loads ? 11. The perimeter of one garden is 1222 inches, the perimeter of another is 2162 inches. What is the length of the longest piece of string which may be used to compare their perimeters ? 12. I wish to distribute 480 oranges and 1184 apples between some school children, giving an equal number of oranges or apples to each child. What is the greatest number of each fruit a child can have ? 13. What is the greatest weight which can be taken an exact number of times from 15 cwt. 2 qrs. 16 Ibs. and 10 cwt. 8 Ibs. without leaving any remainder ? 14. Two companies of soldiers numbering respectively 1632 and 3072 are to be arranged in parallel rows with the same number of men in each. What is the greatest number to a row so as to admit of this being done, and how many rows will there be? 15. What is the greatest amount which can be paid an exact number of times from each of the sums, 4. 15^., ,26. $s. & and ^13. 6.?. without leaving any remainder? 16. There are four heaps of balls, A, B, C and D, numbering respectively 1071, 567, 2583, 1449. They are to be re-arranged in groups so as to have the same number in each group, and without mixing balls from different heaps. What is the greatest number which can be put to a group, and how many such groups will there be ? LEAST COMMON MULTIPLE 191 39. LEAST COMMON MULTIPLE. Suppose we wish to add the three vulgar fractions ^, f and T 4 3-. Each fraction must be changed into an equivalent fraction, and these three equivalent fractions must have the same denominator, and it is clearly important that this Common Denominator should be the smallest possible. Thus, in the case cited, we change T %, f and T 4 T into fj, | and J. The sum is therefore fj; and 60 is the least number which is a multiple of 12, and 4, and 15; in other words, their Least Common Multiple. In the same way we would proceed in adding or sub- tracting other fractions. EXAMPLE. To find the Least Common Multiple of 240, 128, 225. Begin by splitting each number up into its prime factors thus : 128 = 8 X l6= 2 3 X 2 3 X 2 =2 7 . 225 = 25x 9 = 5 2 x3 2 . The multiple must therefore contain a 2, repeated 4 times so far as is necessary for 240 to be a factor ; but 128 contains 2 repeated as a multiple 7 times, and therefore the Lowest Common Multiple (frequently represented by L.C.M.) must contain 2 7 as a factor. Similarly, it must contain 5 2 and 3 2 , and its value therefore = 5- x 3 2 x 2 7 = 28800. 192 A MODERN ARITHMETIC EXAMPLE. To find the Least Common Multiple of ^ x y 3 x 2 4 and 5 3 x y 2 x 2 and 3 3 x 5 5 x y 1 . The L.C.M. of 3 2 x y 3 x 2 4 , and 5 3 x y 2 x 2 1 , and 3 3 x 5 5 x y 1 will be 3 3 x y 3 x 2 4 x 5 5 , in which it will be noted that (1) Each prime factor occurring in the given numbers, occurs in the L.C.M. (2) The power associated with that factor in the L.C.M., is the highest power associated with that factor in any of the given numbers. We may express these rules, using symbols instead of numbers. Suppose one number when split up into its prime factors - and that a second number = a x x b Y x c* x d w ,, where X is greater than x> y .r, Z ,, ,, z. Then the L.C.M. = a x x b Y x c* x e u x d w . EXERCISES LIII. a. Mental or Oral. State or write down the L.C.M. of the following in factor form, without evaluating^ 1. 2 2 x 3, 3 2 x 2. 2. 3 5 x 5 3 , 3 4 x 5 4 . 3. 2 4 x3X5 2 , 2x32, 2 3 x 5 3 . 4. ii 2 x2 4 x 5 3 , n x 2*x 3 3 , 5 x 3x2. 5. 7 4 x 3 1 x 2 3 , 7 1 x 5 4 x 2 3 , 7 1 x 5 1 x 3 7 , 5 x 3 2 x 2 2 . 6. i3X4 2 X2 1 , 4 x X2 2 x i3 2 . 7. 8 1 XQ, 9 2 x 8 3 x 2 1 , 3 2 x 9x4. 8. 5 x io 2 x 4 2 , 4 2 x 10 x 5 2 . 9. 7 x 8, 4 x 7, 8 x 4, 3 x 6. 10, 9 2 x 5, 5 x 4 2 , io x 3. General Method. Occasionally it may be difficult to split up each term into its prime factors. In these cases, and in these cases only, the Least Common Multiple is obtained by successive applications of the principles involved in Greatest Common Measure. LEAST COMMON MULTIPLE 193 EXAMPLE. To find the L.C.M. 0^1391 and 1819. The G.C.M. of 1391 and 1819 can be found in the ordinary way to be 107. By division we see that 1391 = 107 x 13 1819= 107 x 17 And the L.C.M. .'. = 107 x 13 x 17 = 23647. And generally, suppose A and B are any two numbers of which c is the G.C.M. Then A contains c as one factor ; let a contain the others, so that A = a x c. Then B contains c as one factor ; let b contain the others, so that B = b x c\ and as there are no factors common to a and b, the L.C.M. = ax&x.c, of which a common form is a x c x b x c . AB , t.6. . . C C It we wish to find the L.C.M. of three or more numbers A, B, C, D, we find the L.C.M. of A, B, say E, then L.C.M. of E and C, say F, and L.C.M. of F and D, say G. G will be the required L.C.M. of A, B, C, D. EXERCISES LIII. b. Least Common Multiple and Greatest Common Measure. 1. Find the Least Common Multiple of 3 4 x 5 2 x 2 4 and 3 2 x 5 3 x 2 x 7. 2. Find the Least Common Multiple of 7 2 x 2 C x 3, 7 x 3 2 x 2, 5 x 1 1 x 3 3 . 3. Find the Least Common Multiple of (a) 65, 78, 26. (&) 192, 256 and 600. (c) in, 12, 37. (d) 4, 14, 1 8, 24. J.M.A. N 194 A MODERN ARITHMETIC 4. Express as fractions with their denominators equal and as small as possible : () i, *, i, ! <*) & M, <&, A- (0 M, M-, ?, I- (O A, ft, *l, T**- (') T%, i I, tt- 5. Arrange in descending order of magnitude the following fractions : () f^, m, iff- (*) fi> ?*> 42, H- (0 *, m /oV 6. What is the least sum of money which can be paid exactly with an exact number of florins and crowns ? 7. What is the least distance that can be measured exactly by rods 4 ft. and 6 ft. long ? 8. A vessel completely full of water can be emptied by filling from it a beaker of 54 c.c. content, pouring the water away from the beaker and repeating the process. This can also be done with a beaker of 36 c.c. content. Find the least possible volume of the vessel. 9. A fire hearth can be exactly covered by a number of 6 in. square tiles and of 8 inch square tiles. What is the least sized hearth which will admit of this, and how many tiles of each size will be needed ? 10. The floor of a recess may be covered by an exact number of bricks 9" by 3", or square tiles of 8" side, or slabs I ft. square. What is the least area which will admit of this, and how many of each kind will be required ? Evaluate the following : 11- sV-fV 12. 13. &-*+& 14. 15. H-H+if 16. 17. How much is the largest of the following fractions greater than the least, |, ^Jg, if, & ? 18. By how much must the larger of the following fractions be diminished to be equal to the smaller, || EXERCISES 195 19. Find the least number which is an exact multiple of 51 and of 68. 20. A series of numbers is such that each number on being divided by 6 leaves a remainder i, on being divided by 8 leaves a remainder 3 ; show that the numbers in the series increase by 24 ; find the smallest number of the series. 21. A scale graduated in twelfths of an inch is placed with its edge against that of another scale graduated in sixteenths of an inch. The first graduation on each scale is at its end, and the scales are both 2 feet long ; where will the graduations coincide ? 22. Find the smallest number which has a remainder 2 when divided by 126 and also by 432. 23. Find the smallest number which has a remainder 3 when divided by each of the numbers, 75, 24, 40, 15. 24. What is the smallest number which when increased by 5 is divisible by 28, 36, 63 and 108 ? 25. The children of a school may be arranged in classes of 24, 25, or 15, with none over. What is the least number of scholars admitting of this ? 26. When a hogshead of sugar was packed in 2 qr. packets there were 12 Ib. over, and when packed into 48 Ib. packets there were also 12 Ib. over. Find the least weight of sugar to allow this being done. 27. What is the least sum of money which can be paid using one kind only of any of the present silver coins of the kingdom ? 28. A sum of money is to be arranged in piles containing ;i. I2s. and so that there is a single is. over, and also into piles of ^3 and again a single is. over. What is the smallest amount with which this can be done ? 29. A number lies between 1 500 and 2000 ; the number is known to be exactly divisible both by 102 and 36 ; find the number. 30. Find the smallest number lying between 1000 and 2000 which is divisible by 35, 112 and also by 280. 196 A MODERN ARITHMETIC 31. Three men in walking take paces of 32", 30", and 28" respectively. They start together. What is the smallest distance walked when each man shall have reached it by an exact number of paces ? 32. What is the smallest sum in English money which can be paid either by an exact number of francs (each <)\d.) or by an exact number of shillings, and how many francs would there be in this sum ? 33. Each of two rolling wheels has a pin on its circumference which strikes a bell as the wheel turns. Given that the circum- ferences are 54 in. and 198 in., and that each pin strikes its bell at the same instant in starting, how many times will the bells be struck simultaneously in going a distance of half a mile ? 34. On dividing 740 by a certain number the remainder is 2, but on dividing 987 by the same number the remainder is 3 ; find the greatest value of the number. 35. A number divides both 1728 and 1296; find the number, knowing that it lies between 50 and 80. 36. A cubical block is made up of bricks each 28 cm. long, 16 cm. wide and 14 cm. thick, all laid the same way ; find the smallest possible number of bricks in the block. 37. Two scales are placed with their edges alongside one another. In one of the scales the divisions are 0-05 inch apart, in the other scale the divisions are 0-045 mcn apart. What is the smallest distance between divisions exactly opposite one another ? DEFINITION. A number is said to be a perfect square when it can be split up into two equal factors : thus 49 is a perfect square, since 49 = 7 x 7 = 7 2 . If the number consists of 3 equal factors, the number is a perfect cube, and so on. 38. Find the smallest numbers, the product of which with (a) 1008, (d) 1 1025, (<;) 7936 will be a perfect square. EXERCISES 197 39. Find the smallest number which is a perfect square, and contains 6336 as a factor. 40. Find the smallest number containing in the same way as in Ex. 39, (a) 1728, (&) 31212. 41. Find the least number which is a perfect cube, and which contains (a) 1875 as a factor, also which contains (b) 432, (c) 8575, (d) 7986 as factors. 42. On dividing 21892 by 12, by 15, and by 35, the remainders are 6, 7 and 17 respectively. What is the number nearest to 21892, and less than it, which would leave the same remainders? How many numbers less than 21892 are there which would have these same remainders? 43. What is the greatest factor of 2475 which is also a perfect square ? 44. Write down a series of numbers, less than 50, each of which when divided by 3 leaves a remainder I ; write down a series of numbers, also less than 50, each of which when divided by 7 leaves a remainder 6. Note the numbers common to the two series, and then write down a series formed by the 10 smallest numbers, such that when divided by 3 the remainder is i, and when divided by 7 the remainder is 6. 40. PROPORTION. The principle involved is best understood by considering a number of examples. EXAMPLE. A grocer sells 4 Ib. of tea for 9.?.; what will be the price of $ Ib. of tea ? Here it is clear that (1) If no tea is sold no money would be received, and if no money be received no tea would be sold. (2) For each Ib. of tea sold, presumably the same amount of money would be received. I 9 8 A MODERN ARITHMETIC We reason, therefore, as follows : The selling price of 4 Ib. of tea is 95-. i Ib. of tea is J the price of 4 Ib., i.e. | of 9-r. The selling price of 3 Ib. of tea is 3 times the cost of i Ib., i.e. J of 9-y., i.e. 6s. qd. .'. 3 Ib. of tea would cost 6s. 9^. We should generally shorten the work in practice as follows : Selling price of 4 Ib. is 95. 3 Ib. is J of 95-., i.e. 6s. <)d. EXAMPLE. A man buys 12 cwt. of copper for 4$ ; how much could he buy for ^56 ? Again we satisfy ourselves that (1) If no money be paid no copper could be obtained, and if no copper be received no money would be expended. (2) For each expended, the same amount of copper would be received (if the amount of money expended be very large, this might not be the case). We reason then as before : ^"48 buys 1 2 cwt. of copper. 56 jgX 12 cwt. of copper, i.e. - cwt, i.e. 14 cwt. ' ;5 6 is tne buying price of 14 cwt. of copper. Care must be taken to see if the principle involved can really be applied. Consider the following case : The distances through which a body falls vertically from rest are nearly as follows : In the first second, 1 6 feet. 2 seconds, 64 feet. ,, ,, 3 seconds, 144 feet. 4 seconds, 256 feet. Incorrect results would therefore be obtained if, knowing simply the distance fallen through in 3 seconds, we tried to calculate the distance the body would fall through in other times by reasoning as follows : A body falls through 144 feet in 3 seconds. i foot in T f T sec. or }% sec. ,, 1 6 feet in T X F x 16, i.e. in \ sec. PROPORTION 199 According to this calculation, a body falls 16 feet in ^ sec., whereas it takes i sec. to fall that distance in reality. The question has to be asked, whether, to each second added to the time, there corresponds always the same amount to be added to the distance, and also if the distance is nothing when the time is nothing. EXAMPLE. The weight of a sheet of iron 6 ft. square is 4-86 Ib.; what would be the weight of a similar sheet 3'5//. square ? Here (1) The weight becomes indefinitely small as the length of the side of the square becomes indefinitely small ; but it cannot be said (2) That each increase or decrease of a foot in the length of the side gives the same increase or decrease in the weight ; and .'. the weight of the sheet, 3-5 ft. sq., is not ^~ x 4-87 Ib. We could say, however, that (2 A) Each square foot increase or decrease in the area of the sheet means the same increase or decrease in the weight ; and put the question in this way : 36 sq. ft, i.e. (6 x 6), of the iron weigh 4-86 Ib. ; what will be the weight of 12-25 (** 3*5 x 3'5) sc l- ft- of iron? 36 sq. ft. weigh 4-86 Ib. 12-25 -(12-25) (0-135) lb - = 1-65375 Ib. EXAMPLE. A salesman's salary consists partly of a fixed amount and partly of a certain sum for every worth of goods sold. If his salary be ^150 when he sells ^1000 worth of goods, what will his salary be when ^2000 worth of goods are sold? Here, it cannot be said that 1. When no goods are sold, no salary is received; although we can say, 2. For each increase in the value of the goods sold, the salesman's salary increases by the same amount. The question cannot, in fact, be solved on the data above. 200 A MODERN ARITHMETIC Suppose, however, we knew that his salary was ;i8o when the value of the goods sold was ^"1500. The question could now be put differently. If the increase in his salary be ^30 for ^500 increase in the value of the goods sold, what will be the increase in his salary for 1000 increase in the value of the goods sold ? Increase in goods sold to the value ^500 gives ,30 increase of salary. Increase in goods sold to the value ^1000 gives -Vinr x 3 increase of salary, i.e. 60. His salary is therefore 60 greater when he sells ^2000 worth of goods than when he sells only ^1000 worth, and therefore is EXAMPLE. Water consists of 1 6 parts by weight of oxygen to every 2 parts by weight of hydrogen. How much hydrogen can be obtained by decomposing 12-474 grams of water ~1 Here, out of every (16 + 2) grams of water, 2 grams are hydrogen. .*. out of every 12-474 grams of water - 'J^IA x 2 are hydrogen. l8 .'. amount of hydrogen = -^-^ gr. = 1-386 gr. EXAMPLE. How much oxygen will be used up in burning 2 '34 grams of carbon, assuming that in the carbonic acid gas produced there are 12 parts by weight of carbon to every 32 parts by weight of oxygen ? 1 2 gr. of carbon unite with 3 2 gr. of oxygen. 2-34 gr. ~^ x 32 gr. of oxygen. .'. amount of oxygen = ^-^x32 grams = 2 ' 34 ' - grams = 8xo-78 grams 12 3 = 6-24 grams. PROPORTION 201 EXAMPLE. In a Railway the working expenses per mile per annum were as follows : >> Maintenance, - 139 Locomotives and carriages, - - 390 Traffic charges, - 223 Rates and duty, - 49 Miscellaneous, 150 If on the line the total working expenses for the year were 2\<)6%i,find the expenditure under each of the above heads. The total working expenses per mile = ;(i39 + 390 + 223 + 49 + 150) = ^951 ; .'. out of a total of .951 the 'Maintenance' amounts to ^139. Out of a total of ,219681 the 'Maintenance' amounts Out of a total of ^2 19681 the ' Maintenance ' = 231x^139 = ,32109 So out of a total of ^219681 the cost of Locomotives and carriages = 231 x ^390= ,90090 Traffic charges - =231x^223= 51513 Rates and duty - - = 23ix ,49= ,11319 Miscellaneous - =231x^150= ,34650 Total ^21968 1 EXAMPLE. Four people A, B, C and D put ^3000, ^4500, ^"2000 and 2 5 oo into a business concern; the total profits available for division in the ratio of the share capital is ^"3600. Find the profit made by A, B, C and D. Here on ^(3000 + 4500 + 2000 + 2500) the profit is ^3600; .'. on ^3000 the profit is -$$ x ^3600 = ^900 = A's share. on ^45 T 4 ?o% x x36oo = ;i35o = B's on ^2000 T \o oo x ^3 6o = 6o = c ' s on ,2500 T % 5 o7o x ^"3600 = 750 = D's 202 A MODERN ARITHMETIC EXERCISES LIV. 1. If loo kilos, be the French equivalent of 220 lb., what will be the French equivalent of 55 lb.? 2. What also is the English equivalent of 350 kilos.? 3. If 220 gallons be the same as 1000 litres, how many litres will there be in 33 gallons ? 4. How many gallons will there be in 45 litres ? 5. If a train can describe 88 ft. in i sec., how long will it take to describe i mile (5280 ft.)? 6. A train goes at the rate of 45 miles per hour ; how many feet does it describe per second ? 7. If a train describes a journey of 125 miles in 2 hours and 10 minutes, how long would it (moving at the same speed) take to describe a journey of 92 miles ? 8. A steamer moves at the rate of 300 knots per day ; how many hours will it take to describe 175 knots? 9. The cost of 35 oranges being 2s. 4d., how many can be bought for 35-. 4^/. ? 10. If a stream pours 18,000 cubic feet of water into the sea in 12 minutes, how much will it pour into the sea in 35 minutes? 11. If 4800 cubic feet of coal gas cost 12^., what will be the cost of 2500 cubic feet? 12. The weight of 9 square feet of sheet lead, ^ of an inch thick, being 300 lb., what will be the weight of a roof made of such sheet lead, the area being 1800 square feet ? 13. If 4 lb. of tea cost 6s. 6d., how much will 15 lb. cost ? 14. How many lb. of apples can be bought for i. 5^. 6d. if 12 lb. cost is. $d. ? 15. If 14 yards of cloth cost 2. 55-., what will be the length of a piece of similar cloth, value ,4 ? 16. Assuming that, as a rough calculation, the price of a sailing ship is proportional to its tonnage, find the value of a ship of EXERCISES 203 3420 tons displacement, if a ship of 1728 tons displacement cost ^"4800. 17. If the weight of a sheet of brass, 3 ft. long and 2 ft. broad, be 60 lb., what will be the area of a similar sheet of brass weighing looi Ibs. ? 18. If a cube of iron of 42 inch edge weighs 11 tons, what will be the weight of a block whose volume is 24-5 cubic feet? 19. In transmitting power by belting, one wheel makes 420 revolutions for every 300 made by a second ; how many revolutions does the second make while the first makes 280 ? 20. If 84 lb. of sugar could be bought for 1 5.?. 2^1, what would be the cost of 60 lb. of sugar ? 21. How many lb. of sugar could be bought for Js. jd. if 84 lb. cost 1 5-y. 2d. ? 22. If one million cubic inches of air weigh 46-55 lb., what is the weight of i cubic foot (1728 cubic inches)? 23. If one million cubic inches of air weigh 46-55 lb., how many cubic inches will weigh 14-4 lb.? (State your answer to the nearest hundred.) 24. If the weight of a piece of copper wire 4000 ft. long be 130-88 lb., what will be the weight of a piece of similar wire 125 ft. long? 25. What length of a piece of wire similar to that in Ex. 24 will have a weight of 11-452 lb. ? 26. If the cost of carrying a ship's load be the same for each ton carried, calculate the cost of carrying a load of 1440 tons when the cost of carrying 1800 tons is ^2250. 27. If the wages paid to navvies for driving a tunnel be cal- culated at a certain amount for each yard that the tunnel is driven, what will be the wages paid when a length of 18 yards is driven, knowing that the wages paid for a length of 24 yards was 25.4*.? 28. If the cost of covering with linoleum the floor of a room 14 feet square be ^3. 13^. 6d., how much will it cost to cover the floor of a room 16 ft. square with the same kind of linoleum ? 204 A MODERN ARITHMETIC 29. If the cost of painting a cubical block of 8 ft. edge be I2J., what will be the cost of painting a cubical block of 6 ft. edge ? 30. The freight of a cargo being calculated according to its tonnage, what is the weight of a cargo whose freight is ,860, if a cargo of 4200 tons could be sent for ^3010? 31. If a rod of iron 4-2 square inches in section can sustain a pull equal to the weight of 121-8 tons, what pull could a rod of 6 square inches sustain ? 32. If a steel wire, section y 1 ^ of a square inch, can just hold a load of 3-25 tons without breaking, what is the section of a wire which can just hold a load of 2 tons ? 33. If the weight of 25 feet of common lead piping of inch bore be 400 lb., what is the length of a coil of similar piping weighing i ton ? 34. The weight of an anchor and 120 ft. of cable is 2088 lb., the weight of the anchor and 210 ft. of cable is 2898 lb. ; what is the weight of the anchor ? 35. A jug containing 258 cubic inches of a liquid weighs (with contents) 16 lb. ; if 126 cubic inches of the liquid be poured away, the weight is only 12 lb. I oz. ; what is the weight of the jug ? 36. If a man can bore 78 inches per day through granite and 286 inches per day through limestone, what depth of a hole could a man bore in granite during the time that he bores 231 inches in limestone ? 37. If in a town of 2268 people the average daily consumption of water is 6000 cubic feet, what, at the same rate, would be the average consumption in a town of 2079 inhabitants ? 38. If 432 lb. of water can be changed into steam by burning 56 lb. of coke, what weight of coke must be burnt in order to evaporate 729 lb. of water ? 39. In producing coke from coal, 1540 lb. of coke are obtained from 2240 lb. of coal ; how much coal must be employed in the production of 5665 lb. of coke ? EXERCISES 205 EXERCISES LV. Practical. [Apparatus: Paper. Compass. Scales. Wooden Cylinders. Fine Thread. Pins.'] Measurement of Lengths of Curved Lines. 1. Draw a circle 3 cm. radius. (a) Set the dividers to a distance of say 4 cm. ; mark it round as far as it goes ; thus, from A to B, B to C, C to D, measure oft" the remaining distance DA, and find out the distance A, B, C, D, A. This is clearly less than the distance along the circum- ference. (ft) Repeat the measurements, with the dividers set to 3 cm., 2 cm., I cm., 6mm., 5 mm., 4 mm., 3 mm. You observe that the distance round increases as the distance to which you set the dividers decreases, but that when the distance is small all the results become practically the same, and that a curved line may be regarded as made up of a very great number of indefinitely short straight lines. 2. Measure with your dividers the circumferences of circles with the following radii : () 3 inches, 2 inches, i inch. (b) 6 cm., 5 cm., 4 cm., 3 cm., 2 cm., i cm. 3. Repeat the measurements in Ex. 2, using a fine piece of thread instead of the dividers. Start at any mark ; put one end of the thread upon it, previously having knotted the thread, and hold it there with the nail of the first finger of your left hand. Make the thread coincide as nearly as you can with a small part of the curve, and place the nail of the first finger of your right hand upon it. Now release your left- hand finger and carefully place it at the point where your right- hand finger is held ; then, using your right hand, go on to make 206 A MODERN ARITHMETIC some more of the string exactly coincide with another small length of curve. Repeat this until you have completed the whole curve. FIG. 29. Measurement of a curved line with thread. Measure on your scale the length of thread from the knot to the end. 4. Repeat the measurements of the circle in Ex. 2 by sticking a number of pins at close intervals along the circle ; wrap cotton round these pins, and when the cotton overlaps cut it ; measure the cut piece of cotton. 5. Wrap a strip of paper closely round a wooden cylinder, and make a small hole with a pin at a place where the paper overlaps. Unroll the paper and measure the distance round the cylinder. FIG. 30. Measurement of the circumference of a cylinder. 6. From the results of your measurements, see if the length of the circumference divided by the length of the diameter is always the same. The multiple that the circumference is of the diameter is usually denoted by the symbol IT. PERCENTAGES 207 41. PERCENTAGES. EXAMPLE. A particular kind of gun-metal alloy is composed of 10 per cent, tin and 90 per cent, copper. How much tin is there in 3 tons of this gun metal? Here, by the term 10 per cent, is meant 10 parts out of every 100, so that the amount of tin present is 10 hundredths or o- 10 of 3 tons = 6 cwt., and similarly the amount of copper present is 90 hundredths or 0-90 of 3 tons = 54 cwt., and it is clear that per cent, is used after a number to express a fraction. EXAMPLE. The. profit made by a tradesman is 15 per cent, on the selling price. What is the profit made on goods sold for $. i2s. 6</.? The first part of the question means that The profit = T Yo- of the selling price = 0-15 of $. i2s. 6d. = 0-15 of ^5-625- = ^"0-84375 = 165. lold. EXAMPLE. A commission agent obtains a commission of 12. 55. on selling goods to the value of ^150 ; at what rate per cent, does he earn commission ? Here the commission on ,150 is 12. 5$., i.e. 12^. = .! x 4 i 2 | 5 /~bi = ;&Tr- .'. commission is earned at the rate of 8J per cent. EXAMPLE. Type-metal is made of varying composition, ranging from (a) i part antimony to 3 parts lead, up to (b) i part antimony to 7 parts lead. Express each composition as a percentage composition. Here we wish to find the amounts of antimony and lead respectively in each 100 parts of alloy. 2 o8 A MODERN ARITHMETIC (a) In (i + 3) parts of type-metal 3 parts are lead ; in 100 3 x ! parts are lead, = 75 In ( i + 3) parts of type-metal i part is antimony ; .'. in TOO i +i^- parts are antimony, .i.e. 25 The composition of type-metal (a) is therefore 75 per cent, lead, 25 per cent, antimony. (b) In 100 parts of type-metal - x 7 parts are lead, = 87*5 55 55 IOO and in x i parts are antimony, = 12-5 ,, The composition of type-metal (b) is therefore 87-5 per cent, lead, 12-5 per cent, antimony. N.B. The symbol % is used to express per cent. EXAMPLE. A tradesman allows his customers 4 % off their bills when settled within one month of the purchase of the goods ; a certain customer has 4^. 7 \d. returned to him in this way ; what was the amount of his bill? On a bill of ^100, the amount allowed off is ^4; the bill is iJS. times the amount allowed off, and in the example the bill is ^ x 4*. 7^. = ^5. i 5 s. i\d. EXAMPLE. By selling goods at 4. i$s. 4^., a tradesman experienced a loss of 1 2 % on the cost price ; what should be the selling price ', so as to effect a gain of 16 % on the cost price ? Suppose the goods had originally cost ^100 ; then the corresponding selling price would have been ;( 100-12), i.e. ^88; whereas to effect a gain of 16% he should have sold at ^(100 + 16), i.e. 116. .'. the required price is !$-/- times the actual selling price = 29 x 4^ = 6. 5 s PROPORTION AND PERCENTAGE 209 EXERCISES LVI. Where no other data are given, the ratio of the circumference of a circle to its diameter may be assumed as 3-14 : i. 1. The diameter of a halfpenny is I inch, its circumference is 3-14 inches approximately; what is the circumference of a penny its diameter being known to be 1-1875 inches? 2. The driving wheel of a locomotive is 5 ft. 6 in. in diameter ; find its circumference, knowing that the circumference of its trailing wheel is 9-42 ft. and the diameter 3 feet ? 3. If a driving wheel makes 336 revolutions in a journey of 1-25 miles, how many revolutions would the wheel make in a journey of 3 miles ? 4. A carriage wheel is 1-26 metres in diameter and it makes 1 60 revolutions per minute ; what is the speed of the carriage in metres per minute ? 5. One horse on a round-about is at a distance of 12 feet from the central axis, another horse is at a distance of 15 ft. If the inner horse describes a distance of 840 feet in a certain time, how many feet will the outer horse have described in the same time ? 6. One toothed wheel of 2 feet diameter drives another of 3 inches radius ; the second wheel has a third of 2' 6" diameter mounted on the same axis, and the third turns a fourth wheel of 2 inches radius. If a point on the circumference of the first wheel is moving at the rate of 4-4 ft. per sec., at what speed is a tooth on the last wheel moving? 7. An endless strap passes round two pulleys, one being of 2 ft. 3 in. diameter, the other of 7 inches diameter. The large pulley makes 5 revs, per second. What is the speed of the strap and how many revolutions does the smaller pulley make in 28 seconds ? 8. A cage is raised from the bottom of a mine 157 fathoms deep by means of a cable and a winding drum 10 feet in diameter ; J.M.A. O 210 A MODERN ARITHMETIC how many revolutions will the drum make in raising the cage from the bottom of the mine to the top ? 9. The diameter of the earth being assumed to be 8000 miles, find the number of statute miles in 60 geographical miles or knots. (60 knots form the ^ part of the circumference of the equator. Assume the ratio of circumference to diameter to be 3-141 in this and the next example.) 10. Find the distance between two points on the parallel of latitude 60, but differing in longitude by 24-3 sees., the radius of the circle on the earth at latitude 60 being 2000 miles. (The complete parallel would correspond to 360 x 60 x 60 sees.) 11. A mass of wood 16 Ib. in weight displaces 0-256 cubic ft. of water ; how much water will be displaced by a portion of the wood 7-3 Ib. in weight ? 12. If the displacement of a ship of 3200 tons be 112000 cubic feet, how much less water would be displaced if the ship was lightened to the extent of 480 tons ? 13. Chalk consists of 56 parts lime to 44 parts carbon dioxide. On strongly heating the chalk the carbon dioxide is evolved and the lime remains. How much lime can thus be obtained from 3 tons of chalk ? 14. The percentage composition of anthracite coal is as follows : 92-56 parts carbon. 3-33 hydrogen. 2 -53 oxygen. 1-58 ash. How much pure carbon is there in 13 tons 4 cwt. of anthracite coal? 15. Coal gas is obtained, together with other products, by distilling coal. The following is a yield obtained from a ton of Newcastle coal : Coal gas, ----- 9500 cubic feet. f coke, - - - 1 500 Ib. By-productsJ tar, 70 Ib. [ ammoniacal liquor, 80 Ib. PROPORTION AND PERCENTAGE 211 How much of such coal must be treated when the by-products weigh 2310 Ib. ? Find also what gas, coke, and tar will be produced. 16. Britannia metal consists of 140 parts by weight of tin, 3 parts copper and 9 parts of antimony. Find the amount of tin necessary to make a can of Britannia metal, the weight of the can being 2-337 kilo. 17. Pure Epsom salts, without the water associated with it, contains 24 parts magnesium ^ 32 sulphur j- by weight. 64 oxygen Find the amount of magnesium contained in 12-3 grams of Epsom salts. 18. Marine glue is made by heating gently i part by weight of india-rubber and 12 of naphtha, and adding 20 parts by weight of powdered shellac. How much naphtha will be required to make 1-87 Ib. of marine glue ? 19. The receipts per mile per annum on a Railway are made up as follows : / Passengers, 1511. Goods, 1575. If the total receipts be ,351804, what are the receipts from passengers, what from goods ? 20. The cost of running a locomotive, per train mile, is as follows : j General charges, 0-4. Running expenses, 5-5. Repairs, etc., 3-3. Find the cost under each of the above heads during a period when the expenditure altogether was ,2300. 21. The subscription to the ' Games Club ' in a school of 200 boys is 6^. 6d. per term, the money being expended on Cricket, Football, and Athletic Sports. If Cricket receives 2s. for every gd. 212 A MODERN ARITHMETIC received by Football, and for every 6d. received by Sports, find the expenditure under each head per year ? 22. Three persons have 3 shares, 4 shares, and 7 shares re- spectively in a business concern, the total number of shares held by other people being 42. The profits amount to ,1400 ; what do the three persons severally receive ? 23. The perimeter of a given triangle (i.e. the total length of the three sides) is 49 inches, the triangle being of the same shape as another triangle whose sides are 17-22 inches, 17-91 inches and 8-62 inches respectively ; what are the lengths of the sides of the given triangle ? 24. The gold coinage of this country is 22 carats fine, i.e. contains 22 parts of pure gold and 2 parts alloy. If the weight of a sovereign be 123-274 grains, what weight of pure gold was required for the coinage of 7500700 sovereigns ? 25. The rateable values of three adjoining parishes are severally ,15,345. loj-., ,18,924. los. and ,12,330. A rate of ^2425 is made upon them. What proportion has each parish to contribute ? 26. Three towns have respectively 14,001, 11,628 and 4845 males capable of serving as soldiers. A company of 482 men has to be raised from the three towns. Apportion the number of men to be found by each town. 27. A, B, C and D join to buy a ship valued at ,7400. A takes 9 shares, B 11 shares, C 17 shares and D 27 shares. Apportion the purchase money between them. 28. Two farmers rent a meadow for ,2. 15^.; one puts in 37 cattle and the other 43. What amount of rent should each pay ? 29. A specimen of brass consists of 13 parts of tin to every 112 parts of copper. Find its percentage composition. 30. The percentage composition of a mortar is as follows : 32-5 lime. 67-5 river sand. i oo-o mortar. How much sand is there in 4 ton 3 cwt. of the mortar? PROPORTION AND PERCENTAGE 213 31. Find the amount per cent, of nitre-glycerine in dynamite consisting of 73 parts nitro-glycerine, 2 parts coal dust and 290 parts of clay. 32. Standard gold consists of 22 parts of pure gold to 2 parts of copper. Find the percentage composition of standard gold, i.e. find, out of every 100 parts of standard gold, how much pure gold there will be and how much copper. (State the answer true to the second decimal place.) 33. Tough brass, suitable for engine work, is made by alloying together 15 parts of tin, 100 parts of copper and 15 parts of zinc. Find the percentage composition of the brass. 34. Pewter consists of 100 parts of tin and 17 parts of antimony. Find its percentage composition. 35. Blue vitriol, or sulphate of copper, is a chemical compound, made up of 63 parts copper, 32 parts sulphur, 64 parts oxygen. Find its percentage composition. 36. ' Stone lime ' is made up of i bushel of lime to 3 of sand and 1-5 bushels of water. Find its percentage composition by volume. 37. Average limestone consists of 112 grs. of carbonate of lime to 9 of clay and 29 of sand. Find its percentage composition. Find out also how much pure carbonate of lime could be obtained from 20 ton 5 cwt. of limestone. 38. Dynamite used for blasting is made up of 287 parts of clay, 2 parts of coal dust and 75 parts of nitro-glycerine. Find its per- centage composition. What weight of nitro-glycerine will there be in 32-2 kilo, of dynamite ? 39. Find the percentage composition of ordinary caustic soda which contains 23 parts by weight of sodium. 1 6 parts by weight of oxygen, i part by weight of hydrogen. 40. The copper ore generally employed in smelting consists of about 30% iron and 13% of copper. How much copper could be obtained by smelting 14 tons of the ore ? 214 A MODERN ARITHMETIC - 2 4j 41. A man buys a horse at a fair, selling it again later for thereby making a profit of 10 % on the buying price. What did he pay for the horse ? 42. If by selling goods at a profit of 8 % on the buying price, a tradesman makes a profit of 3. 6s. &, what was the buying price of the goods ? 43. Express as convenient vulgar fractions the following per- centages : 3i%, 33* %, 25%, i2fc%. 44. What percentages are the equivalents of the following vulgar fractions: J, |, <fo, 5*0 > 4> ? 45. Most rocks are porous and can absorb water. The following table gives data as to this property for a number of rocks. Supply the figures which are missing in the table (true to 3 signi- ficant figures). Geological Formation. Locality. Vol. of water absorbed by 100 vol. of rock. Gallons of water absorbed per cub. ft. of rock. (a) (A) Old Red Sandstone Bristol Clifton IO-30 0-642 1-018 \ u l to Freestone Cheltenham 35-29 46. Complete the following table : TABLE OF CRIMES AND OFFENCES IN COUNTIES IN 1880. (*) County. Population. Census 1881. Number committed for Trial. Number per 1000. Bedford Berks 149,000 2l8,OOO 94 0-47 to w Bucks Cambridge - 176,000 185,000 86 56 47. A very rich kind of limestone contains 95-89 % of carbonate of lime ; assuming that in burning this limestone into lime 56 % PROPORTION AND PERCENTAGE 215 of the carbonate yields lime, what quantity of this limestone is required to yield i ton of lime? 48. The wages of mechanics in a district are reduced from an average of 2. 8s. gd. per week to ^2. 5^. 6d. After a time they have an advance of 7^ % on their later wages. Find the reduction per cent, in the first instance, and the average weekly wages in the second case. 49. By selling a book for 35-. 9<f., ten per cent, of the cost is loss ; at what price should it be sold to make a gain of 10 % ? 50. A certain publishing firm gives a discount of 33^ % off the published price, counts 25 books as 24, and gives a further dis- count of 2^ % on the reduced price. What does the publisher get for a book the published price of which is 16.?. ? 51. A market woman buys 200 eggs at five for threepence and an equal number at eight for sixpence. She sells the whole at a uniform price and gains 33^ % on her outlay. What is the selling price per score ? 52. A house was sold for ,560, and thereby a loss of 9 per cent, was made. If it had been sold for ^650, what would then have been the loss or gain per cent. ? 53. A trading concern in I yr. made a profit of ^1875 on a turn- over of ,23,500 ; in the 2nd year a loss of ^710 was sustained on a turnover of ^22,360 ; while in the 3rd year a profit of ^935 was made on a turnover of ,24,140. What was the average profit per cent, on the 3 years' turnover ? 54. A mass of ore of 3 ton 2 cwt. 2 qr. loses weight during smelting to the extent of 17 cwt. Express the loss as a percentage. 55. If, when goods are sold at ^26. 7^., a profit of 24 % is made on the buying price, what did the goods cost ? 56. What is the cost price of goods on which, when sold at 3. Ss. gain, a profit is made of 15 % on the selling price ? 57. The population of a town increases in two years from 23,280 to 23,862 ; find the gain % during that time. 2i6 A MODERN ARITHMETIC 58. In calculating the area of a rectangle, the real length of which is 47 ft. 3 in. and breadth 28 ft. 7 in., the computer writes 49 ft. 6 in. instead of 47 ft. 3 in. ; what percentage error does he make? 59. A merchant allows a customer 5 % discount on a bill, and receives a net amount ^182. 17^. 6d. ; what was the bill? 60. Eight trams contain 7 tons 5 cwt. of coal from a seam A. When the coal is screened it yields 3 tons 5 cwt. i qr. of large coal ; and an equal weight from a seam B when similarly screened .yields 4 tons 14 cwt. i qr. of small. Compare the per- centages of large coal in each seam. 61. A man, whose gross income is ,300, pays income tax at is. in the on the excess of his income over ,160 ; what percentage is the amount paid of his gross income? 62. A rod of brass is 425 cm. long at 15 C., and at 100 C. its length is 425-065025 cm. ; find the percentage gain in length. 63. 35 % of an estate of 2 sq. miles 24 ac. is set apart for building purposes ; how much is left ? 64. The value of land near a town increases by 33.^ % during a certain period. Find the increase in the value of an estate of 660 acres, originally worth ,45 an acre. 65. When a metal is cast in a mould the portion of the casting near the surface has generally to be chiselled off. If an amount of 4800 gr. is removed in this way, and the remaining portion be 200 kilo., what percentage of the original casting has been taken away ? EXERCISES LVII.a. Graphic. [Apparatus : Scales. Mathematical Instruments. Squared Paper. ~\ 1. On the squared centimetre paper mark off two straight lines at right angles to one another, letter them Qx and Oy (Fig. 31), O being at the meeting place of the lines. PROPORTION 217 Along Qx set off a distance ON = io cm.; measure upwards a line NP=9 cm. Join OP. At the end of each cm. in ON, starting from O, say at the points NI, N 2 , N 3 , etc., draw the verticals NjP^ N 2 P 2 , N 3 P 3 , ..., etc., and test the lengths with regard to the two fundamental relations. 1. Do they alter by equal amounts as the horizontal distances ON 15 etc., alter by constant amounts ? 2. Do they be- come indefinitely small as ON n etc., ~O N, N 2 N 3 become indefinitely Fie. 3 i. small ? Test also OP with regard to ON, and OP with regard to NP. 2. Draw a number of other lines and test in the same way. 3. Obtain points P 15 P 2 , ...as before by setting off ON n ON 2 , ON 3 , etc., along O.r, and NjPj, N 2 P 2 , etc., parallel to Oy, where ONj, etc., and N^, etc., are given by the following tables (values given in cm.). Join the points P 1} P 2 , etc. ON, N,P, ON 2 N 2 P 2 ON 3 N,P, ON 4 N 4 P 4 ON, N 5 P 5 ON 6 NP 6 i 1-2 2 2-4 3 3-6 4 4-8 5 6 6 7-2 0-8 0-7 1-6 0-8 1-4 2-4 2-1 3-2 2-8 4 3-5 4-8 4-2 0-4 I 2 1-2 3 1-6 4 2-0 5 2-4 6 1-2 0-6 2 I 2-8 1-4 3-6 1-8 4-4 2-2 5-2 2-6 In each case notice that N^. etc., increase by equal amounts for equal increases in ON 15 etc., and that they vanish together. 218 A MODERN ARITHMETIC 4. Obtain points representing pairs of quantities, where, although equal increases occur in one for equal increases in the other, they do not vanish together. (a) (6) (<-} (d\ ONj N;P! ON 2 N 2 P. 2 ON 3 N 3 P 3 ON 4 N 4 P 4 ON 5 N 5 P 5 ON 6 N 6 P 6 o 2-2 i 4 2 5-8 3 7-6 4 9-4 5 1 1 -2 o 0-6 i 1-2 2 1-8 3 2-4 4 3-o 5 3-6 2 I 3 1-4 4 1-8 5 2-2 6 2-6 7 3 I 2 2 3 3 4 4 5 5 6 6 7 5. Obtain points where the quantities concerned are not so related that equal changes in the one correspond to equal changes in the other. ONj N lPl ON 2 N 3 P 2 ON 3 N 3 P 3 ON 4 N 4 P 4 ON 5 N 5 P 5 ON.lN 6 P 6 o o i 0-2 2 0-8 3 1-8 4 3-2 5 5 o i o-3 2 1-2 3 2-7 4 4-8 5 7-5 o o. 5 0-2 I 1-6 1-2 5-4 2 12-8 (a) W EXERCISES LVII.b. 1. If the rail of a railroad remains straight, and if in moving along it for a distance of 460 yards horizontally, a vertical rise of 2-3 ft. is experienced, after what horizontal distance will the rise be 3ft.? 2. In driving a slant of 440 yds. through a seam of coal, there is a uniform dip of 4-5 inches in each yard. How much lower is the bottom of the slant than the top ? 3. If the rise in a straight railroad be i vertical in 114 hori- zontal, what vertical rise will occur in a horizontal displacement of 3249ft.? PROPORTION 219 4. A yacht makes a straight tack in a certain direction, and afterwards finds it has moved 4-08 miles to the east and 3-89 miles to the north. How much further north will it have gone when it is 11-73 to the east of its original position the same tack being pursued all the time? 5. A yacht sails for 5 22 miles along a tack, making 3-1 miles east for every 4-65 miles measured in the direction of the tack ; it then changes its course, now making 3 miles west for every 8 miles along the new course ; after pursuing this new tack for 5-8 miles, how far will it have moved east of its original position ? 6. A road rises at the rate of 12-32 yards for each mile of road for a distance of 872 yards from a place A ; the inclination then increases to 23-6 yards per mile until B is arrived at, distant I mile 91 yards from A, measured along the road. How much higher is B than A ? 7. After going down the 'incline' in a mine for 183 yards, a point is reached 43-26 yards below the surface. The total length of the incline is 280-6 yards. What is the depth of the bottom of the incline below the surface ? 8. A ladder consists of 40 steps, each 32 cm. apart. One end of it rests on the ground, the other against a wall at the height of 9-32 metres. A man ascends the ladder ; how high will he be when he has gone up 28 steps ? 9. When a pile of dry sand is made, the sand begins to slip once the steepness is greater than that corresponding to a rise of 71-4 cm. for each horizontal distance of 91 cm. Find the greatest height of a sand castle (without patting), the base of which is 213-2 cm. in diameter. 10. A wall is 8-262 feet high. Dry sand is thrown against it until it forms a slope just reaching to the top of the wall. Find the least distance of the bottom of the slope from the wall, (Data as in Exercise 9.) 220 A MODERN ARITHMETIC Scale Drawing. Very few objects can be drawn of exactly the actual size ; they have to be shown either reduced or enlarged to a certain scale. Thus, a large scale Ordnance Map may be drawn to a scale of 6 inches to the mile, where a real distance of i mile is represented by a distance of 6 inches on the map. A map of England may be shown to a scale of 100 miles to the inch. The fraction that the distance on the map is of the distance it represents is known as the Eepresentative Fraction, and is frequently represented by the letters R.F. A scale 6 inches to the mile could therefore be called a scale where R.F. = 6/63360, i.e. R.F. = 1/10560. EXAMPLE. TJie real distance between two towns we may call them A and B is known to be 232 miles ; on a map they appear to be 3-16 inches apart. C and D, two other towns, appear to be 4-74 inches apart on the map. What is the real distance between C and D ? A distance of 3-16" on the map represents an actual distance of 232 miles. .'. a distance of 4-74" on the map represents an actual distance of 232X tl| miles 232 X * ., = =2 a miles 2 = 348 miles. The distance from C to D is therefore 348 miles. EXAMPLE. Find the Representative Fraction on a map where a distance of 7 fur. 8 ch. is represented by a line 5-148 inches long. Here 7 fur. 8 ch. = 78 ch. = 78 x 22 yds. = 78 x 22 x 36 inches. The fraction which represents the ratio of the distance on the map to the actual length represented is : 5-148 = 5-148 = i 78x22x36 12x5148 12000' In other words, the R.F. = 1/12000. SCALE DRAWING 221 EXAMPLE. A scale is constructed of 6 inches to the mile ; a comparative scale of kilometres has also to be constructed. What would be the length of line on the new scale which represents a distance of 2-31 kilometres ? By a Comparative Scale, with regard to a given scale, is meant one with the same Representative Fraction but in which different units are employed ; and therefore, by using the two scales, a distance may be read off in different units, say in miles or in kilometres, etc. A scale of 6 in. to the mile has ^f^ir for its R.F. (p. 220). A new scale is required by means of which distances can be read off, not in miles but in kilometres. Since the R.F. is still ^^y, a distance of 2-31 kilometres is represented by 2-31 x 1000 x 100 = I05 6o - Cm ' I-06 EXAMPLE. In testing the. magnifying power of a /ens, it is found that 6 divisions on a millimetre scale appear enlarged and to coincide with divisions on a centimetre scale 5-6 cm. apart. What is the real radius of a circle which appears to be of 1-26 inches radius ? Here the lens magnifies times, o-o or the magnification is ^ diameters. o-o The apparent radius = The real radius multiplied by the magnification. The real radius of the circle, therefore, is 1-26 x r inch 5'6 0-756 . = -~ inches 5-6 = 0-135 inch. 222 A MODERN ARITHMETIC EXERCISES LVIII. Practical. {Apparatus : Mathematical Instruments. Squared Paper.] 1. In the accompanying plan of a room the real distance between the points H, M is 23 feet 9 inches. Measure off the distance on K C D . P M FIG. 32. the plan between the points H, M, and then, measuring also the dis- tances on the plan between the points mentioned below, calculate by proportion the real distance between them. (a) A to B, (b) C to D, (c) L to M, (d) K to M, (*) P to R, (/) P to Q, (g) R to Q. 2. Draw a new plan in which HM is represented by a line 3 inches long, the other lines being altered in proportion. 3. The accompanying map (Fig. 33) is without a scale, but it is known that the distance from Stockton to Doncaster is 61 miles. Construct a diagonal scale for use with the map ; use it to find the distances between (a) York and Lincoln, (b) Doncaster and Skegness, (c) Sheffield and Scarboro. 4. The distance from London to Exeter is 150 miles; find, by means of measurements on the accompanying map (Fig. 34), the distance from London to (a) Bristol, (b} Reading, (c) Gloucester, (d) Birmingham, (e) Norwich, (/) Derby, (g) Northampton. SCALE DRAWING 223 Durham Stockton rboro FIG. 34. 224 A MODERN ARITHMETIC 5. The sea passage from Newhaven to Dieppe is 75 miles ; rind the length of the passage from (a) Dover to Calais, (b) Folkestone to Boulogne. ais Boulogne Dieppe FIG. 35. EXERCISES LIX. Miscellaneous. 1. Define a prime number. When are numbers said to be prime to one another? Write down all the prime numbers below 60. Find the prime factors of 14348907. 2. If a man pays 14. 13^. $d. as income tax at *jd. in the ,, what is his full income before paying the tax ? 3. Find the cost of 4217 articles at 8^. ^d. each. 4. Subtract j| of ,16. 2s. 4^. from 0-0125 of .1626. i$s., and find by what decimal the result must be multiplied to produce i. 6s. i\d. REVISION EXERCISES 225 5. ^u. i8j. yd. is paid for a piece of work done by A, B and C. A has worked 42 hours, B 77 hours and C 48 hours. If C does in 2 hours as much as A or B does in 3 hours, what should each man receive ? 6. Convert ff into a decimal (5 places), and 0-00003125 into a vulgar fraction. 7. What fraction, the denominator of which is 580, equals f ? 8. A rectangular block of stone is 5 yds. 7 in. long, 12 ft. 6 in. broad and 3 ft. 9 in. thick. Find how many cubic inches it contains. 9. What weight must be added to f of of half a cwt. to make it equal to -fa of 3f quarters avoirdupois ? 10. Gunpowder is composed of fa sulphur, fa charcoal, f nitre by weight. How many Ibs. of each will be required to make 1 1 tons of gunpowder? 11. The sum of three fractions is 2- The first is ^ and the second is f . What is the third ? 12. How much is (a) i5j + of 7f ; (b) 0-8125 of 2 tons 4 cwt. ; (c) 78-125x40-96? 13. If 3 fires burn 65 cwt. of coal in 26 days, in how many days will the fires burn 4 tons ? 14. The population of a town fell from 9600 in 1881 to 8544 in 1891 ; how much per cent, was the decrease? 15. Of 21 tons 5 cwt. of coal, 19 per cent, is wasted- What weight is wasted ? 16. A sum of money was divided among 3 men. If the first had }, the second f and the third 38^., what did the money amount to ? 17. What is a prime number. Give one example to show that two numbers which are prime to each other are not necessarily prime numbers. Resolve 17424 into prime factors, and hence (or otherwise) find the side of a square field which contains 3 acres 2904 square yards, and reduce it to the decimal of a mile. J.M.A. P 226 A MODERN ARITHMETIC 18. What is the value of 3 Ib. 7 02/5 dwt. 6 grs. of silver at 3,?. 4d. per oz. ? 19. Find the lowest whole number which will convert 0-02369 into a finite decimal, by subtracting the decimal from one a hundred times as great and studying the result. 20. If a person who invests ^9000 obtains 4 per cent, per annum on two-thirds of it and 5 per cent, per annum on the rest, what income do the investments yield him, and what percentage does he obtain for his money ? 21. Find the value of 0-384375 of ^5. \f 20 TO ^-*> T rA -* 1 :T.*i /A\ 1Z 'J^~/'.) 23. (a] How much must I add to i-Jf to make 2 T J(j? and (b} find a simple fraction which is equal to T \^- 24. A can dig a garden in 4^ days, while B can dig the same in 6 days. How long will A and B together take to dig half the garden ? 25. A man buys 87 horses for ,1740. He sells 25 at 7 per cent, profit on the cost price, and 40 at 1 2^ per cent, profit ; he loses 3 by disease, and sells the rest at cost price. How much does he gain ? 26. If 40 men earn ,36. 8s. in 8 days, how many men will, at the same rate of wage, earn ^40. igs. in 8 days ? 27. At an election a candidate polled 6 per cent, less than his opponent, who obtained 2160 votes ; how many votes were given for the unsuccessful candidate ? 28. What is the debt, 71 per cent, of which is ^301. 15^. ? 29. If a cubic foot contains 6| gallons, how many gallons of water are there on a field of 2^ acres, when it is flooded to a uniform depth of an inch ? REVISION EXERCISES 227 30. The carriage of 18 cwt. 3 qrs. for a distance of 14 mis. 2 fur. amounts to g. What should the carriage of 5 tons for 19 mis. amount to at the same rate ? 31. Take the sum of 2 and ^ from the sum of 6, 4f and ^. 32. A man's income is ^459. 17$. 6d. per year, and of this he saves ,15. 6s. *]d. ; what percentage of his income is this? 33. Find all the prime numbers that divide both 1287 and 1144 without any remainders. 34. A tradesman marks his goods 20 per cent, above cost price, but makes a reduction of 10 per cent, on the marked price for ready money ; what is his gain per cent, on ready money trans- actions ? 35. A and B ride a race of 31 miles on bicycles. The driving wheel of A's machine makes 3410 revolutions per hour, and has a circumference of 168 inches ; that of B makes 3520 revolutions per hour, and has a circumference of 162 inches : which will win, and by how much ? 36. If a person who invests ^5000 obtains 4^ per cent, per annum on three-fourths of it, and 3^ per cent, per annum on the rest, what income do the investments yield him ? 37. If i6| tons of coal, bought for 7. $s. 2<, are carried 324 miles at a further charge of \d. per ton per mile, what is the total cost of the" coal per ton to the purchaser ? 38. Find the value of 0-0625 of i. 4$. + 0-02 5 of 5 guineas + 1-125 of los - &/., then reduce the sum to the fraction of 3. 4^. 6d. 39. A, B and C rent a field between them for 2.0. los. A puts in 25 sheep for 8 months, B 24 sheep for 9 months and C 40 sheep for 6 months. How much should each have to pay ? 40. One gallon of spirit which contains 1 1 per cent, of water is added to 3 gallons containing 7 per cent, of water, and to this mixture half a gallon of water is added. Find the percentage of water in the mixture. 22 8 A MODERN ARITHMETIC 41. Find (by fractions) a number such that, if one twenty-ninth part of it be taken away, the remainder is 377. 42. Add together ^Ssi and 1 3& S - 43. Find the square root of the least integer that is a common multiple of i||f, 22J-J, and i^. 44. A watch is offered for sale at ^5. 15^. od. : and, if that price is reduced by 5 per cent., the dealer who is selling it will still make 9^ per cent, profit on the cost price ; how much did the watch cost him ? 45. If to every quart of milk a milkman adds half-a-pint of water, and if he sells the mixture at ihe same price per quart as he gave for the milk alone, what is his profit per cent. ? 46. Explain the meaning of a prime factor. Write down all the prime factors common to 3003 and 8778. 47. Divide (by decimals) 72 thousandths by 25, and also by nine ten-thousandths. 48. Taking the values of zinc and copper to be 17. 14^. od. and ,73. 15.?. od. per ton respectively, find how much of each metal there will be in a mass compounded of the two, which weighs 14 cwt., and is worth ,23. \2s. od. ? 49. A man loses 30 per cent, of his property and has ,1400 left ; how much had he at first ? 50. Explain the rule for finding the Least Common Multiple. Find the smallest sum of which i^s. yd., i. us. 6d. and ^3. i$s. are exact parts. 51. Eight bells, which toll at intervals of i, 2, 3, 4, 5, 6, 7, 8 seconds respectively, begin tolling all simultaneously with a clock striking ; how many hours must elapse before they all toll simultaneously again with the clock striking ? The clock is supposed to strike at the hour only. 52. Three railway tickets, a ist, a 2nd and half a third class were purchased for i6s. \o\d. The ist class ticket costs if times as much as the 2nd, and the 2nd class i^ times as much as a whole REVISION EXERCISES 229 3rd class ticket. The distance travelled was 45 miles. Find the cost of each ticket and the rate per mile for each class. 53. What number is that from which if y 7 ^ T^ be deducted, and to the remainder 4- be added, the sum will be 3$f ? J 4 54. If with sales amounting to ^iooo per month a tradesman gains ^100 in 7 months, what sales would he have to make in order to gain ^60. los. in 1 1 months ? 55. Distribute ,630 among A, B, C and D ; so that B and C may together have half what A and D have together, that B's money may be f of C's, and D's money of A's. 56. Find the square root of the Least Common Multiple of 7^, 19^- and 2o. 57. Express 6-72 oz. as the decimal of i\ cwt., and explain the process by which you obtain your answer. 58. A population of 84,700 people loses during a pestilence 5929 persons ; what is the loss per cent. ? 59. If air contains 23-01 % of its weight of oxygen, and if a cubic ft. of air weighs 1-23 oz., find to the nearest ounce the weight of oxygen in a room 18 ft. wide, 27 ft. 4 in. long and 12 ft. high. 60. Show that the difference between any improper fraction and unity is always greater than the difference between unity and the reciprocal of the fraction. [ Take any improper fraction which you like to select, and show that the proposition is true for that fraction j then extend your reasoning to improper fractions generally^ 61. The residue of an estate was left to be divided between three persons, A, B, C, in such proportion that A's share was to be to B's share as 4:5, and B's share to C's as 9 : 16 ; the residue realised ^2415. How much was each person entitled to? 62. If you invest a sum of money in such a way that on one-third of it you gain 3 per cent., on one-fifth you lose 4 per cent., and on the remainder gain 6 per cent. ; what average rate per cent, do you make on the whole sum invested ? 2 3 o A MODERN ARITHMETIC 63. The value of a certain house in 1880 has increased 35 per cent, since 1877. The house was rated in 1877 at two-thirds of its value, and in 1880 it is rated at three-fifths of its value, the rate in the remaining the same. Compare the rate paid in 1877 with that paid in 1880. 64. What is the greatest number that will divide 1035 and 1665 without remainders ? 65. After deducting income tax at 8d. in the from a sum of money, ^508. qs. ^d. is left ; what is the sum ? 66. Find the difference between lof and if, and then divide it by the sum of 7\ and 3^. 67. Divide i9 T y* by 4yf T- 68. I have to collect 21 for a society. In the ist week I collected 4. Js. 6d, and in the 2nd week 8. i6s. $d. What fraction of the whole sum was collected, and what fraction remains uncollected ? 69. A druggist buys a certain commodity by avoirdupois weight and sells it by troy weight. The buying price is ^d. per oz., and he sells at 6d. per oz. What is his gain per cent, on the buying price ? 70. What is the rateable value of a parish if a rate of y. $\d. in the produces a sum of ,14,352. is. 8d. ? 71. A person sold a horse at a loss of 20 per cent, on the cost price ; if he had received 10 more for it, he would have gained 10 per cent, on the cost price. Find the cost of the horse. 72. A man leaves ,32,818 to be divided among his four sons in the proportion of the fractions , f, and f. Find the share of each. 73. What is the value of 4^ per cent, of ^1350 ? 74. Goods sold for ^207. 14^. <^d. gave a profit of three-eighths of their cost. Find their cost. 75. F'ind the dividend on ^4146. \2s. 6d. at nj. 8d. in the . 76. How often is each of the quantities 19^, IQ, and 2i contained in their Least Common Multiple ? 77. Reduce ^i. 17,?. \%d. to the fraction of ^22. 5^. 6d. REVISION EXERCISES 231 78. Reduce igV x 3$ +6 fa to a decimal. 79. If the cost of a piece of cloth, 28 inches wide, be 8 guineas, what should be paid for a piece twice as long and 25 inches wide? 80. If a man buys eggs at 10 for a shilling, and sells them at 8 for a shilling, what rate per cent, profit does he make on his outlay ? 81. After spending jj- of my money at A, then of the remainder at B, and next ^ of what still remained at C, I had $s. f)\d. over. How much had I at first ? 82. When 2 cwt. I qr. 2 Ib. of sugar are taken away from a ton seven times, what decimal of a ton still remains, and what is the value of the remainder at ^20-8 per ton ? 83. Simplify each of the following: ()ff|, (*)||,(')(2f +ifH&. 84. A heap of cannon balls can be arranged in groups of 34, 51, or 170, leaving 10 over from each group. What is the least number of balls the heap can contain ? 85. At noon I had completed of my journey, at 2.15 I had completed f of it, and had travelled 63 miles since noon. Find (a) the total length of my journey, (&) the uniform rate of travelling, (c) the times of starting and arriving. 86. Find the cost of 389 pairs of boots at ,3. Js. 6d. for 4 pairs. 87. State the rule for the multiplication of fractions, and give the reason for it. Which is the greatest of the fractions /g, ^, g ? 88. A person, after spending half of his money, gives away 17.?. 6d. He has then 17^. 6rf. left. How much had he at first? 89. Divide 21. ?s. 6d. between a man, a woman, and a boy, so that for every 5^. the man gets, the boy may get 35-., and for every 5-r. the woman gets, the boy may get 3^. gd. 90. A salary of 600 rupees a month (12 months to the year) was once worth ^720 a year; but, owing to the depreciation of silver, it is now reduced by 31^ per cent. What is the present value in English money of a rupee ? 232 A MODERN ARITHMETIC 91 Simplify 4?-*of4Vk+64. 4 f -18^8 -$ of if ' 92. A man pays a tenth of his income in rates and taxes and a twelfth in insurances, then he has left 49 2 - 1 3 S - l ^- What was his original income ? 93. A person directed in his will that \ of his property (,38,657. 6s. 6d.) should go to A, to B, \ to C and to D. Show that these directions could not be carried out. Show how it should be divided so that the shares may have to one another the ratio he intended. 94. A builder (A) sells to an agent (B) a house for ,4860, by which he loses 19 per cent, on the cost price ; B disposes of it to C at a price which would have given A 17 per cent, on the cost price profit. Find B's total gain. 95. The children of a school can be arranged either in classes of 32, 48 or 40 with none over. What is the least number of scholars which will permit this, and how many classes will there be of each number ? 96. If gold is worth ^3. 17.9. 6d. an ounce, and a nugget of gold weighing i Ib. 6 oz. 15 dwt. contains 15% of gold, what is the value of the gold ? 97. If 3^ tons be carried a certain distance for 7. 15^. 2<f., how much should be charged for 95 cwt. for the same distance ? 98. A, B and C have a pasture. A puts 500 sheep on it for 4 months, B 600 for 3 months and C 1200 for 2 months. If the whole rent be ,279, how much should each pay ? 99. The sum of three fractions is f. The first was #\, the second J$. What was the third ? 100. Find the prime factors of 495, 429 and 195, and employ them to find the square root of the product of these numbers. 101. A farmer bought a flock of sheep numbering 864 for 1720, and kept them for 6 months at a cost of 14^. <)d. per head : he then sold them at ^3. is. Sd. each. What profit or loss was made? REVISION EXERCISES 233 102. A farmer has 14 milking cows, each giving 2 gallons I pint of milk per milking. They are milked morning and evening, and the milk sold at i^d. per pint. What is the value of the milk sold in the month of May ? 103. A man buys 45 bushels of apples (40 Ib. to the bushel) for 6. One third of them is spoilt ; he sells a third of the remainder at id. a Ib. and two thirds at 4^. a bushel. What profit is made, and what fraction of the whole cost is profit? 104. A mother receives 2. los. q\d. weekly, and spends in 5 weeks j. \is. ^\d. How long will she be in saving 105. Three tons of goods were bought at the rate of >\d. per Ib., and the carriage was $d. a Ib., what is gained by selling them at $. 12s. per cwt. ? 106. How many turkeys, each worth i$s. 7|^., are equivalent in value to 1 502 geese at 95-. $d. each ? 107. If the product of $ and 2} be added to the sum of 2 if and 3 f \, by how much will the result differ from 100? 108. A man spends 65 % of his income and saves ,434. What is the amount of his income, and what sum does he spend ? 109. The population of a town on Jan. i, 1901, was 64,000. It increased 5 per cent, in each year ; what was the population on Jan. i, 1904? 110. An English sovereign contains 113 grains pure gold: 1 5'432 grains make i gramme; a 20 franc piece contains 5-8 grammes of pure gold. What number of francs should i be equal to ? 111. A merchant sells an article at an advance of 12 per cent, on the cost price ; how much per cent, of the selling price is profit ? 112. The gross rental of a farm is ^520. The owner pays income- tax at jd. in the ,, and then 5 % of the remainder to the collector of the rent. How much will the owner himself receive? 234 A MODERN ARITHMETIC 113. An army of 145,600 men loses 12 % of its number through disease and 15 % of the remainder in battle. How many men are left ? 114. A solicitor charges a certain percentage for negotiating a mortgage of ^1450 on a house. His bill is ,23, of which 15 guineas are law costs. What does he charge per cent, for negotiating ? 115. If 12% of an army be disabled in the first month of a campaign, and 15 % of the remainder in the second month, and 10 % of the latter remainder in the third month, it then numbers 33,660. What was the original number of men, and the number disabled each month. lit}. In coining shillings at the Mint, a composition is used consisting of 37 parts of silver and 3 parts of copper. The price of the silver is $s. zd. an ounce, and the weight of a shilling is T 2 T of an ounce. What is the value of the pure silver in the shilling? 117. 40 Ibs. troy of standard gold can be coined into 1869 sovereigns, the proportion of pure gold to alloy in standard gold being 22 to 2. What weight of pure gold is there in a sovereign ? 118. A, B, C agree to pay their Hotel Bill in the proportion of 4:5:6. A pays the first day's bill, which is 2. los. \od. ; B the 2nd day's, which is ^3. 125. 6d. ; and C the 3rd, which is ^3. 17^. How must they settle their accounts ? 119. Three men A, B, C have a grazing farm and pay ^42. I2s. rent : A puts in 90 oxen, B puts in 96 and C puts in 98. What proportion of the rent should each pay ? 120. A farmer bought 4 horses and 7 cows for ^238, and the price of 2 horses was equal to the price of 5 cows. How much did he give for each ? 121. A works for 6 days of 8 hours a day ; B works for 5 hrs. on the first day, and on each of the 5 subsequent days i hr. longer than on the preceding day. A does as much in 4 hrs. as B does in REVISION EXERCISES 235 5 hrs. The total sum paid to A and B for wages for the work is 2. 2s. How much should each receive ? 122. A legacy of ^2420 is left to three persons in such propor- tions that after the payment of a legacy duty of 15 per cent, the first receives twice as much as the second, the second three times as much as the third. What are their respective shares ? 123. A bankrupt, whose estate is worth ,698, has 4 creditors, A, B, C and D. He owes A ,270, B ^300, C ^150 and D /ioo. The legal expenses of his bankruptcy are ^83. How much will each creditor receive ? 124. A person bought an estate and subsequently sold it for 62 5 less than he gave for it, thereby losing ij per cent, on the cost price. What should he have received in order to gain 12^ per cent, on the cost price ? 125. If a grocer gains 10 per cent, by selling tea at 2s. ^d. per lb., what will he gain per cent, on the buying price by selling it at 2s. yd. per lb. ? 126. If 4 per cent, be lost by selling silk at los. per yard, at what price per yard should it be sold in order to gain 5 per cent. ? 127. Find the value of f of 7^. 6^.4-0-625 of los. -ff of ys. id., and express the result as the decimal of ^4. 128. Multiply ,15 by 7-8498, giving the answer true to the nearest penny. 129. If 45 sheep cost 20, how much will 27 cost ? 130. What dividend will be paid on a debt of ,906. los. at 15-y. id. in the ? 131. What percentage of ^1625 is ,52. i6s. 3^.? 132. What is the lowest number divisible exactly by 33, 22, 55 and 30 ? 236 A MODERN ARITHMETIC 42. CALCULATIONS AND APPROXIMATIONS. In actual business transactions answers and results rarely work out exactly, and the figures are only approximations. EXAMPLE. A man asks his butcher for a piece of beef to weigh about 10 Ib. A piece is cut off, and on weighing is found to be 9 Ib. 13 oz.or nearer 9 Ib. 13 oz. than 9 Ib. 12 oz. The butcher calculates the price at say 1 1 d. per Ib. on the assumption that the weight is 9 Ib. 13 oz. What is the cost of the beef? The price is therefore nd. x 9^1 = 85. n-\^d., but 8s. iiy|</. is nearer 95-. than Ss. n^d., because \^d. is nearer to id., i.e. \^d.) than to a halfpenny, i.e. -%d. Accordingly, the price charged is 95. NOTE. It would probably be found, however, that the butcher would charge gs. for anything over Ss. n^d. and less than gs., whether nearer to qs. or not. EXAMPLE. Find the cost of 115 tons 8 cwt. 25 Ib. of coal at I2S. 6d. per ton. Coal would be weighed to the nearest quarter probably, and the price calculated accordingly. Thus, when sold at 1 2 s. 6d. per ton, an amount of coal 115 tons 8 cwt. 2 5 Ib. in reality would have only been weighed to the nearest quarter, and the price calculated on 1 1 5 torfc 8 cwt. i qr. s. d. The cost of the 115 tons would be 1437 . 6 8 cwt. 5-o i qr. 0.2, though really the fourth part of i\d. The total cost therefore would be 14425-. Sd., i.e. 72. 25. 8d. CALCULATIONS AND APPROXIMATIONS 237 EXERCISES LX. a. Mental or Oral. Find true to the nearest \d. the cost of the following : 1. Shoulder of mutton (a) 4 Ib. 10 oz., (b) 5 Ib. 2 oz., (c) 4 Ib. 12 oz., (d) 6 Ib. 3 oz., at yd. per Ib. 2. Sirloin of beef (#) 7 Ib. 12 oz., (<5)4 Ib. 6 oz., (c) 7 Ib. 2 oz., (</) 8 Ib. 9 oz., at i \\d. per Ib. 3. Piece of cheese (a) 5 Ib. 3 oz., (b) 4 Ib. 5 oz., (<:) 6 Ib. 8 oz., (d} 12 Ib. 6 oz., at \\d. per Ib. 4. A length of silk (a) 2 yds. 34 in., (b) 3 yds. 16 in., (c) 4^ yds. long, at 4-y. 3^/. per yd. 5. A piece of cloth 4 yds. 8 in. long, at $s. per yard. EXERCISES LX. b. Find true to the nearest \d. the cost of the following : 1. A truck load of coal, 7 tons 14 cwt. 2 qrs., at 1 1 s. $d. per ton. 2. A mass of 34 cwt. 27 Ib. of copper, at ,63 per ton. (Weighed and calculated true to the nearest Ib.) 3. A length of cloth 3 yds. and an eighth, at 3.?. $d. per yd. 4. A piece of timber 7 ft. 8 in. long, at 8d. per foot run. 5. Two pounds three ounces of tea, at is. 7\d. per Ib. Find the weights of the following masses : 6. 25^ cubic feet of water assuming that i cubic foot of water weighs 62-5 Ib. (State answer true to the nearest Ib.) 7. A brick wall 67^ feet long, 9 inches thick and 7^ feet high. (Assume i cubic foot of brickwork weighs i cwt., and state your answer true to the nearest cwt.) 8. A prism of iron^io ft. 3 in. long and 87 sq. inches in cross section. State your answer true to the nearest quarter, (i cubic foot of iron to be assumed as weighing 450 Ib.) 9. A cable length of 120 fathoms, each foot of which weighs 2 Ib. (State your answer true to the nearest qr.) 238 A MODERN ARITHMETIC 10. The yield of a vein of coal, 2 ft. 9 in. thick, 6 acres in extent. (State your answer in tons true to the nearest hundreds digit, assuming that approximately i cubic yard in the mine yields one ton of coal.) 11. What quantity of coal per acre can be obtained from three seams, severally 2 ft. 3 in., 2 ft. 6 in. and 2 ft. 9 in. thick, assuming the same data as in Ex. 10? (True to the nearest hundred.) 12. The yearly output of a limestone quarry being 23,536 tons, find the yield of lime to the nearest hundred when the stone yields 52 per cent, of lime. 13. What is the cost to the nearest penny of cutting 63 tons 1 1 cwt. 2 qrs. of coal, at is. 1\d. per ton ? 14. Find also the cost to the nearest penny of cutting 95 tons 15 cwt. i qr. if the price per ton be 30 % above that in Ex. 13. 15. Find the cost to the nearest penny of 43! gall, of oil at is. id. per gall. 16. Find the cost of 4 tons 17 cwt. 3 qrs. of pit wood, at 195-. 6d. per ton. 17. What is the value of 134 Ib. 9 oz. of old brass at 6\d. per lb.? 18. What is the value of a copper ingot weighing 2 cwt. 21 lb., at 8^. per lb.? 19. 3000 fire-bricks are assumed to weigh 9 tons. What is the weight of a brick to the nearest lb. ? 20. Find true to a thousand gallons what additional quantity of water is added to a reservoir whose area is 55 acres 1200 sq. yds. by a rainfall of 0-9 inch, assuming 277 c. inches to a gallon. 21. Find true to a ton the yield of hay from a meadow of 5 ac. 3 ro. 25 sq. po., at 4 tons to the acre. 22. Find true to the nearest quarter the yield of wheat from an acreage of 146 acres 2 ro. 16 po., at 3^ quarters to the acre. 23. Find to the nearest cubic yard the quantity of earth removed in forming a ditch 220 yds. long, 8 ft. wide and 3 ft. 3 in. deep. 24. Estimate to the nearest thousand the number of gallons of water pumped from a mine per day by an engine, when it makes CALCULATIONS AND APPROXIMATIONS 239 3 strokes per minute and discharges 19-3 cubic ft. per stroke, assuming I cubic ft. of water to equal 6-25 gall. 25. Find to the nearest acre the area ol a rectangular park 2 m. 600 yds. long and i m. 800 yds. wide. N.B. It should be carefully noted that the degree of approximation varies in each particular case. For example, a tax was made on salaries, investments, etc. The amount taxed was ^608,606,903 ; the tax could be determined at a certain amount in the , fixed exactly to a \d. A variation of \d. in the rate per makes, however, a difference of over ^1,400,000 in the produce from the tax. 26. A town estimates its expenditure for the year as ^26,000, true to the nearest thousand pounds. If the rateable value of the town be ,170,000, find to what degree of accuracy (nearest penny or halfpenny, etc.) the rate per can be fixed. 27. A train is drawn along at a nominal speed of 80 ft. per sec. Ii the speed varies to the extent of i ft. per sec., what difference might it make in the distance described in 12 hours' continuous run ? 28. If, owing to unequal arms, a balance is out by 3-5 Ib. in a cwt., what is the amount of the error when 96 tons 18 cwt. 3 qrs. are weighed by it ? Decimalization of Money. The fact that the money payment in exchange for goods and service is only approximate, is important in its bearing on the use of the decimal system. is. =0-05 of a . i d. = 0-004 1 666... ;, i.e. = 0-00416^". \d. = 0-00104166. ..;, =0-0010416^. So that, frequently, to express a sum of money as a decimal of a would involve the use of recurring decimals, and the decimal system might then be inconvenient for calculation. If the calculations are given, however, to the nearest farthing, and it is noted that ^d. = 0-00104166... and that 240 A MODERN ARITHMETIC ^"0-001=0-96 farthing, it will often be sufficient to state the decimalized sum of money to the nearest figure in the third place of decimals. EXAMPLE. To decimalize a sum of money, say ^14. i2s. ^d. (a) When the methods of ordinary reduction are employed, the working will be as follows : 14. i2S. 7^=^14. i2S. 7-25^=^14. 12-604166. ..s. = ^14-63020833... . Frequently, especially in mental and approximate calculations, it is more convenient to express the pence and farthings as decimals of a , without first expressing them as decimals of a shilling. Thus : i farthing = - = -/V^-P-P^ - 1 = -/" f ^-P-i- 4 - V- 1 \ = ;( J + 'annr) Tinny = >(^ + A) ' 01 j and, therefore, if the pence and farthings be expressed as a number of farthings, and this number be increased by its 24th part (usually this will only be required to the nearest unit), the result will give the pence and farthings as thousandths of i. f Applied to the same example, the working (mental) would be as under : (b) 7 \d. = 29 farthings Increase 29 by its 24th = 0-030^ approx. part, true to the nearest unit, I2J-. = 12(0-05)^ and regard each unit as a = 0-60^. thousandth of . 14. i2S. l\d. = ^14-6 30 approx., but true to 3rd decimal place. More accurately, 14. i 2 s. i\d. =14-629 + ( Y( 0>O2 9)- EXAMPLE. To decimalize ^103. 13*. *\d. 2^d. = 10 farthings = ^0-010 approx. ' _^*/~i.i->Tn\ or accurately, ^13*. = 13 x ^0-05 =^0-650. ' I0 3- I 3- y - 2 2^- = ; I0 3-66o, true to the 3rd decimal place. More accurately, ,103. 13^. 2 \d. =^103-660 + 24 CALCULATIONS AND APPROXIMATIONS 241 EXAMPLE. To decimalize 25. Ss. 17 farthings 17 increased by its 24th part is = 0-0 1 8 1 8, true to the nearest unit. 8.9. = 0-4, .' 25- &s. 4i</. = 25-418 approx. = 25-417 +^~^~ accurately). Conversely, the method of converting sums of money ex- pressed in decimals of , to pounds, shillings and pence, is illustrated by the following example : EXAMPLE. To express ,36-243 as . s. d. Each. 0-05 corresponds to a Is., each 0-001 is a \d. nearly. And 36-243 = 36. 4^. + 43 farthings nearly = 36. 4-r. + \o>\d. nearly. To be more accurate, we should remember that in decimalizing the farthings we take each \d as 0-00 1, and then increase the result by ^ T . Here, therefore, we should diminish the 43 farthings by their 24th part, and thereby obtain a close approximation. .'. since ^jth of 43 = 2 approx. 0-043 = 41 farthings approx. and 0-2 = 4^. _ ' 36-243 = 36- 4* iJ<* Evaluating 36-243 in the ordinary way, 36-243 = 36 + 2o(-243)j. = 36. 4-86*. = 36. 4^. 10-32^. = 36- 4^- ii^- to the nearest farthing. EXAMPLE. To express 6-183 as s - d - 0-183 = 3(0-05) + 0-033 33 reduced by its 24th part = 32 approx. .*. 6-183 = 6. 3- 9 - + 3 2 farthings approx. = 6. 3^. %d. true to the nearest farthing. J.M.A. Q 242 A MODERN ARITHMETIC EXERCISES LXI.a. Decimalize the following, or express in . s. d. (true to the nearest farthing or to 0-001), by the preceding methods, and verify in the manner employed on pp. 240-241 : 1. 26. is. 2\d. 2. 26. is. i\d. 3. 35-121. 4. ,35-108. 5. 98. i8j. i\d. 6. 15. IIA & 7. 83-696. 8. 34-112. 9. ,17-857. EXERCISES LXLb. Mental or Oral. Express as decimals of : 1. 2. 2s. od. 2. 2. y. od. 3. 5. is. od. 4. 5. Ss. od. 5. 3. 17-r- od. 6. 4. I5-J- ort 7 . 7. 102. 13.?. od. 8. 116. 19^-. odl 9. X. 10. Id. 11. i\d. 12. 2\d. 13. ild. 14. 2. or. 8i<f. 15. 12. oj. 9!^. 16. n- ictf- 8^. 17. 12. Ss. 7\d. 18. 12. Qj. 7\d. 19. 12. is. q\d. 20. 101. y. 4%a '. 21. 13. 2J. 1^. 22. 13. 7s. $\d. 23. 14. 6s. \\d. 24. 114- is. ijfl EXERCISES LXLc. Mental or Oral. Express . s. d. as decimals of a ,, or vice versa, in the following : 1. 2. i3j. 6^. 2. 3. 8j. 4^. 3. 101-123. 4. 6. 17^. o^. 5. 3. 1 5 j. o^. 6. 7. i6s. id. 7. 3. 14.?. ij^. 8. 3-287. 9. 6-102. 10. 12-301. 11. 8-293. 12. 16. iSs. 7d. 13. 3. i6j. 4d. 14. 1-726. 15. 2-318. 16. 100. 2s. od. 17. 100. is. lod. 18. 23. i7s. iid. 19. 15. 6.r. Sfrt 7 . 20. 3-167. 21. 2-135- 22. 2. y. 6\d. 23. 7. is. 2\d. 24. 13-006. 25. 8-103. 26. 9. 2s. 7\d. CALCULATIONS AND APPROXIMATIONS 243 Possible Errors in Decimalization. In the method of decimalizing sums of money that has already been explained, it must be noticed that frequently it may not be accurate enough to decimalize to the third place, when the sum of money is expressed in . EXAMPLE. To evaluate the cost of 30,000 articles at 12. 2S. ^\d. each. 12. 2S. 4\d.=\2-i and 19 farthings = ^(12-1 19 + -2^ of 0-019) = ;i2-i2o true to the third decimal place. Taking the cost of 30,000, at this assumed value for the single article, the cost works out = 30,000 x i 2-120 = ,363,600 ; whereas the correct cost = 30,000 x ;i2-i 19 + 30,000 x Y x ^0-019 = ^363,570 + ^^ ' = ^363,570 + ^23. 15*. 93- I 5 S - with an error of & 5 s - d. EXAMPLE. Evaluate the cost of '3600 articles at i. iSs. 7\d. each. 1. iSs. >]\d. = ,1-9 + 30 farthings = ^1-9 + ^0-030+^^^ 3600x^1. 18.?. 7 \d. = 3600 x ;i-93o + ^x 3600x^0-030 = ^694-8 + ^4-5 = ^699. 6s. It does not always follow that the above is the quickest or most advantageous method of evaluating, and it is mentioned here in order to point out the possible error which might creep in when the approximate decimalized value is used. In dividing by numbers greater than i, there is no danger in approximating, since the error gets reduced. 244 A MODERN ARITHMETIC EXERCISES LXII. 1. Evaluate the following : (a) i. 17$. 4</.x(i) 100, (ii) 1000, (iii) 800, (iv) 480. () ^156. 4J. 2f^.-=-(i) ioo, (ii) 700, (iii) 300. (c) ^2892. or. 7^x0) 26o > (") I2 5- 2. In finding the cost of 600,000 articles at i2s. l\d. each, the result is calculated by reducing i2s..7%d. to the decimal of a , true to the third decimal figure. What is the error so produced ? 3. Find the cost of (a) 125 articles at ,3. los. 2d. each. (p) 200 articles at ,4. us. 2\d. each. (c) ioo articles at i. us. o^d. each. (d) 450 articles at ^3. los. <\\d. each. 4. What error would ensue if, in evaluating the following, the approximate values in decimalizing are used before multiplying ? . s. d. (a) 1000 articles at o . 10 . i. (b) 5600 o . 7 . 2\. (c) 84,000 0.1.4. (d) 96,000 i . 3 . oj. 43. PRACTICE. Practice is a simple and convenient method of computation by means of aliquot parts. An aliquot part of a number or quantity is a part which will divide into that number or quantity without remainder. Thus, 2$. 6d. is an aliquot part of i, for it is exactly J- of ^i. The method of finding cost by practice sums is often used by tradesmen, and will be evident from the following worked examples. It will be noted that the process admits of many variations. PRACTICE 245 (a] EXAMPLE. What is the value 4/873 articles at 3. $s. 10^. each? i- * 873 . o . o cost if articles were \ each. 3_ o 3 o 5 J - > (/>.4th the cost at ^i each). 7 J cost if articles were 7 </. each (/>. Jth the cost at 55. each). 3 cost if articles were 3d. each (z>. -^jth the cost at $s. each). L each. |th of 55-. 3d.= 6s. 5 J - 2619 218 27 10 . 18 2875 . 8 . lojcost at 3. 5f. s- d. 873 .0.0 3 of 6*. 2619 . 261 . o . o cost at 18 . o 2880 . 5 18 . o 9. ii each. 6s. 2875. 8. 3. 65. each. \d. each. By subtraction obtain cost More frequently it is advisable to work in decimals. 6d. = T \jth of 55. 3</. = J of 6d. [J^.=J of 3d. 873 3 2619 218-25 21-825 10-9125 5-45675 cost at each. 2875-44425 cost at 3. 5.?. ioj</. each. Cost, therefore, ^2875. 8^. 2 4 6 A MODERN ARITHMETIC EXAMPLE. Find the cost of 547! cwt. at per cwt., using the decimal method. 547-375 cost at * per cwt. 2s. of of 2s. of % 54-7375 18-24583 1-14036 2s. 621-399 (To nearest figure in 3rd place of decimals, since 0-39869 is nearer 1.00-399 than 100-398.) .*. cost = ^62 1. TS. n^d. EXAMPLE. find the value of 237-! tons at o. gs. *]d. tier ton. 237 =cost of 237 tons at i. L = Jth of 4.?. f.=ithof 6d. 59-25 47-4 0-9875 = 5 JJ 6d. 3^. = cost of 237 tons at 9^. . 5</. = ,, ton at ^. 237! tons at gs. EXEECISES LXIILa. Mental or Oral. 1. What aliquot parts of a are the following sums : i or., 6s. &/., sj., 4j., u. 8^., 2J., 2J. 6^., 3^. 4^., r ij lortf., u. 3^. ? 2. State as aliquot parts of a shilling : 6^., 4^., 3^., 2^/., i^/., i^. 3. What aliquot parts of a ton are : 10 cwt, 5 cwt., 4 cwt, 2 cwt, i cwt ? PRACTICE 24? 4. State as aliquot parts of a cwt. : 2 qrs., i qr., 56 Ibs., 28 IDS., 16 Ibs., 14 Ibs., 7 Ibs. 5. What aliquot parts of a quarter are : 14 Ibs., 7 Ibs., 4 Ibs., 3^ Ibs., 2 Ibs., i Ib. ? EXERCISES LXIII.b. Mental or Oral. 1. What aliquot part of a is each of the following ? (a) y. */., (S) 5J ., (c) 2*, (rf) w. 4*, (') u. 3^, (/) 6j. 8^, Qr) 7 R (*) 5<* 2. What aliquot part of loj. is each of the following? (a) 2s. &/., (*) w., (f) sj., (</) w. 3^, (e) 7&, (/) 5^, (jf) & **, (A) iU 3. What aliquot part of 4^. is each of the following ? (a) is. */., (*) 2 j., (f) 8^., (^) 3^-, (0 2^., (/) i^. 4. What aliquot part of 2s. 6d. is each of the following ? (a) !<**, (^) 6d., (c} sd. t (d) tf. t (e) *\<L What amounts should be placed in the following empty brackets ? 5. 2J .3rf.is(*)iof( ),is(J)Jof( ),is(^)iof( ). 6. u.4^.is()tof( ), is(^)iVof( ),is(f)*of( ). 7. i3J.4/iis(a)Jof( ),is(J)Jof( ), is(^)-|-of( ). 8. 2^. is(tf)Joof( ), is^^ofC ),is(f)Jof( ). Find the values of the following articles : 9. (a) 1 5 at is. 4^., () 30 at i j. 4^., (c) 31 at i j. 4^., (d} 29 at i j. $d. 10. () 6 at y. 4d., (b) 18 at 3* 4^., (^) 24 at 3^. 4^., (^) 25 at y. 4^., (^) 23 at 3J. 4d 11. (a) 36 at 4^., (3) 27 at 4^., (r) 333 at 4^., (^f) 48 at 4^., (^) 49 at 4^. 12. (a) 24 at \od., (ff) 48 at iod,, (c) 50 at icW., (^) 46 at icW., 0) 97 at icv/., (/) 121 at \od. 248 A MODERN ARITHMETIC 13. () 32 at 7\d., () 64- at 7^., (<r) 65 at 7^., (</) 96 at 7^., (*) 100 at 7K-> (/) IQ o at 7K 14. (a) 12 at 1*. 8</., (b) 24 at u. 8<, (<:) 26 at u. 8^., (d) 48 at u. 8^., (*) 50 at ;i. u. 8dl, (/) 96 at ^2. u. 8^. 15. (a) 8 at 2s. 6d., (b) 8 at i. 2s. 6d., (c) 16 at 2s. 6d. 9 (d) 15 at ji. 2J. 6</., (e) 48 at ^i. 2J. 6d., (/) 97 at ^i. 2s. 6d. EXERCISES LXIII.C. Calculate the value of the following : 1. 1973 articles at (a) 2d., (6) yt., (c} 4^., (d)6d., (e) 8^., (/) iod. each, using in each instance but i aliquot part. 2. 2375 articles at (a) \\d., (b) i\d., (c) 3^., (d) $, (e) $%d., (/) 7K, (g) 8K, (A) 6f^, (i) 9 ^, (/) ioiW. 3. 584 articles at (a) 2J. 3^., (^) y. 5^., (f) 4*. 8^., (^) 5^. 6^., (e) 6s. icw., (/) 7J. 4</., (^) 8j. 9^., (/^) 9^. 8^., (/) 10.$-. 8^. 4. 3719 articles at (a) 2s. $\d., (b} \y. 8^., fc) 15^. 9^^., (af) i8j. 5^., W i&f. 9^, (/) i8j. Jf-i (^) 19-y- 2^., (^) i9J. 4^- (* Using only i aliquot part.) 5. 6934 articles at (a) i. 12s. 8</., (b) 2. 14*. 7\d., 6. 735 articles at (a) 23. 6s. io^f., (b) ^34. I2J. 9f<, ^42. I5J- 7^-, (^)^53- I7J- 8^1 7. 5931 articles at (a) ^18. <w. i^., (b) 27. os. n%d., (c) ^9- y- i^, (^) ^45- 5-f- 3l^. 8. 7582 articles at (a) 9. y. gd., (b) 10. 12s. 6d, (c) 8. 2s. Sd, (d) $. 8s. 44, (e) 19- 19* 4- 9. 5327 articles at (a) 73. >js. 3^., (b) ^28. 9^. 6|^, (^) ^54. i6j. 4^., (^) ^76. I7J- 10^. 10. 6751 articles at (a) ^3. 3^. 9^., (^) ^5. 6s. 3^., (f) ^8. 2s. 8d., (d) 6. i$s. 4d., (e) 2. ijs. 6d., using only one aliquot part in each case, and then checking your result by using two. PRACTICE 249 EXERCISES LXIILd. Find the cost of the following, giving results true to the nearest penny : 1. 357! yds. at (a) 2s. iod., (b) y. iod., (c) $s. &/., (d) Ss. 6d., (e) i$s. gd., (/) i8j. -id. each. 2. 6i^j articles at (a) 2. 45. $*., (b) 5. 11*. 8^., (c) 9- I4-T- 7d>, (d) 36- 17 s. gd. each. 3. 39SH cwt - at (*) ^3- 5* 4^-1 W ;i- 9-f- 6</., (*) ^5. 13^- *d. t (d) 9. i8j. 6^/. each. 4. 59| miles at (a) o. 4*. \od., (b) 2. Ss. iod., (c) 9. 12s. 5^., (d} 14. 14*. 3|^. per mile. 5. 765! oz. at (a) y. 8^., (b) 3. 175. io\d., (c) $. 45. gd., (d} 8. os. gd. each. 6. 593 tons at (a) i$s. 4^., (b) 2. y. iof^., (c) $. ly. $<i., (d) 9. 14*. 7^d. each. 7. 429 bushels at (a) AS. 6</., (b) o. sj. 8^., (^ ^i. 7.?. 9^., (d) 2. $s. 8^. each. 8. 1573 articles at () 14* 7}d., (b) 3. i$s. 6~d.> (c) 4. 7s. 5^., (d) ? 19^- 2^. each. 9. 201 5 articles at (a) i$s. oj<f., (^) ^4. 2^. 3f</., (<r) ^7. i6j. (d) 3. I2s. 8|</. each. 10. 673! quarters at (a) 4s. 2%d., (b) o. 6s. 4$d., (c) 2. los. (d) $ 17-?. 8 T V- per qr. In the case of Compound Practice the same remarks apply as in Simple Practice (see pp. 245-246). EXERCISES LXIII. e. Mental or Oral. Find the value of : 1. (a) 5 cwt. at los. a ton, (b) 10 cwt. at 3^. 4d. a ton, (c) 4 cwt. at 2. 75. 6d. a ton. 2. (a) i ton 10 cwt. at 9^. 6d. a ton, (b) 2 tons 5 cwt. at iSs. a ton, (c) 70 tons 2 cwt. at 5^. a ton. 3. (a) 1 6 Ibs. at is. gd. a cwt, (b} 14 Ibs. at 2J. 8d. a cwt., (<:) 8 Ibs. at 28^. a cwt. 250 A MODERN ARITHMETIC 4. (a) 8 oz. avoirdupois at is. lod. a lb., (b) 4 oz. at is. $d. a lb.. (c) 2 oz. at lod. a lb. 5. (a) The rent of a stable for 3 months at 7. los. the year, (b) the rent of a house for 6 months at ,11. 15^. od. the year, (f) the rent of an office for 73 days at ^i 80 a year. 6. (a) ii yd. at 3^. 8d. a fur., () 55 yd. at i$s. 6d. a fur., (c) 2 chains at 6.y. 8d. a mile. 7. (a) 4 fur. at ,23. 5^. 6d. a mile, () 9 inches at 2^. 3^. a yd., (*:) 2 yd. 1 3 in. at y. a yd. 8. (a) 4 pints at lod. a gal., () i peck at 3^. 6d. a bush., (c) 4 bush, at 45^. a quarter. 9. (a) 25 packets at 9<^. per 100, (b) 48 pen nibs at is. $d. per gross, (c) 250 bricks at 32^. 6d. per 1000. 10. (a) 12 grains at 2^. per dwt., (b) 144 grains of gold at ,45 perlb. EXERCISES LXIII.f. Find the value of each of the following quantities, giving the answers true to the nearest penny where necessary : 1. 17 cwt. 2 qr. 21 lb. at (a) i. los. 4d., at (b) ^3. 14^. %d. and (c) 7 per cwt. 2. 85 cwt. 3 qr. 14 lb. at (a) 2. i6s., at (b} 3. i?s. and (<:)$. I4s. 4d. per cwt. 3. 160 tons 14 cwt. i qr. at (a) o. us. 8^., at (b) 2. 4s. 2d. and (c} 9. 6s. &d. per ton. 4. 1135 tons 17 cwt. at (a) 8j. yd., (b) i$s. and (c) 3. 5^. iod. per ton. 5. 19 lb. 10 oz. at (a) is. Sd., (b) $s. and (c} 4s. ^d. per lb. 6. 34 miles 6 fur. at (a) 6. 13^. 6^., (b} ^18. i$s. and (c) 47. us. $d. per mile. 7. 84 m. 968 yd. at (a} ^5. I2J-. 6^. and (^) ^24. 6^. 3d. per mile. 8. 13 m. 65 ch. at (rt) 9. los. 4d. and (5) ^8. 14^. 6d. per mile. 9. 47 m. 101 poles at (a) 8. los. and (b} 17. iSs. gd. per mile. MISCELLANEOUS EXERCISES 251 10. 59 yd. 25 in. at (a) is. ^d. per yd., at (b) 2s. -$d. per yd. 11. 74 ac. 3 ro. 27 sq. po. at (a) 2. iSs. 4^., at (b) 4. js. 6d. per ac. 12. i ro. 35 sq. po. at (a) i. 4s. Sd. and (b) 2. los. per ac. 13. 134 ac. 1331 sq. yd. at (a) ^2. 5^. and (b) ,3 per ac. 14. (a} 73 gal. 5 pt. and (b) 28 gal. 3 pt. at U. Sd. per gal. 15. (#) 3689 qr. 3 bush, and (b) 974 qr. 5 bush. 2 pk. at 4s. Sd. per qr. 16. (a) 935 pints at lod. per gal. and (b) 495 bushels at 6s. $d. per quarter. 17. (a) 53 hr, 36 min. labour at \od. per hr. and (b) 4435 min. at 2s. 2d. per hr. 18. (a) 17650 bricks at ,1. 7s. 6d. per 1000 and (b} 860 envelopes at u. 2d. per 100. 19. () 43 Ib. 10 oz. of silver at 5.?. 6d. per oz. and () 1390 grains of gold at ,3. iSs. per oz. EXERCISES LXIV. Miscellaneous. 1. What dividend will be received on ,485. 8^. ^d. at 4-y. 6d. in the? 2. Find the value of 763 mining shares at 155-. 4^d. per share. 3. What is the weight of 546 yards of steel rails at 2 qr. 10 Ib. per yd. ? 4. What will be cost of covering 384 sq. ft. with lead, running 1 1 Ib. per foot super., at J^d. per Ib. ? 5. Find the railway charge on the transit of 750 tons 14 cwt., for 90 miles, at \d. per ton per mile. 6. Find the amount of the rates in the year on an assessment f ,39456. io.y. at 2s. lod. in the for the ist half year, and 2s. Sd. in the 2nd half. 7. What is the income tax on an assessed income of ,1135. 18.?. gd. at the rate of jd. in the ,? 252 A MODERN ARITHMETIC 8. A field of 24 ac. 3 ro. 30. sq. po. yields 4 quarters of wheat per acre. Calculate the value of the wheat at 32$. 6d. per qr. 9. A hay rick measures 1656 cubic ft. ; find its value at 4 guineas a ton when 12 cubic yd. weigh i ton. 10. Find the estimated weight of coal in a taking of 54 ac. 3 ro. 18 sq. po. at 14,520 tons per acre. 11. Copper being sold at 72. $s. 6d. per ton and brass at ,48. los. yd., how much dearer is 6 cwt. 60 Ib. of copper than brass ? 12. Find the distance run by an express train starting at 10.56 a.m. and reaching its destination at 3.45 p.m. the same day, at the rate of 58 miles 800 yd. per hour. 13. What weight of water does a cistern hold when it contains 9 cubic ft. 648 cub. in. ? (i cubic foot of water weighs 62-5 Ibs.) 14. What is the half-year's rent for 3 gardens, each 2 ro. 25 sq. poles, at ^3. 3-y. 4^. per acre per year ? 15. A merchant's debts are ,2375. i8.y. qd., but he is only able to pay a dividend of I2s. lod. in the pound. What is the value of his assets ? 16. The shares of a company are quoted one day at ^32^, and at the end of a week they have fallen to 29^. What difference in value does this represent to a man holding 96 shares ? 17. What is the value of a gold chain weighing i oz. 12 dwt. 1 8 grs. at ,4. 1 6.y. od. per oz. ? 18. What is the yield of a field of 12 ac. 36 sq. po. at the rate of 4 tons 13 cwt. 70 Ib. per acre ? 19. What is the cost of enclosing a square field of 436 yards side with wire rope, running 3 Ib. to the yard, at js. per cwt. ? 20. Find the duty on the following imported articles. (Employ whatever method you think most suitable.) (a) 3752 Ib. of coffee ; duty 2d. per Ib. (b) 1796 gal. of brandy ; duty nj. 4^. per gal. (c) 96,726 cwt. of sugar ; duty 2s. ^d, per cwt. (d) 2674 cwt. of crystallized fruit ; duty 4$. 2d. per cwt. (e) 189 tons 9 cwt. 14 Ib. tea ; duty 6d. per Ib. MISCELLANEOUS EXERCISES 253 (/) 4 tons 6 cwt. 6 Ib. cigars ; duty 6^. per Ib. (g) 36,894 Ib. cigarettes ; duty 4^. lod. per Ib. (Ji) 15 tons 17 cwt. tobacco ; duty 3.?. lod. per Ib. (z) 2 tons 13 cwt. 2 Ib. snuff; duty 43. ^d. per Ib. (f) 178 tons 1 6 cwt. unmanufactured tobacco ; duty $s. 3^. per Ib. (k) 9872 gal. sparkling wine ; duty 2s. 6d. per gal. 21. Find the excise duty on : (a) 198,700 gal. beer ; duty 7$. qd. per 36 gal. (to nearest id.}. (b) 783 tons glucose ; duty 2s. qd. per cwt. (c) 187 cwt. saccharine ; duty is. $d. per ounce. 22. Income Tax being calculated according to the scheme below, find the net income, gross income, or tax, in the following cases : PROPERTY AND INCOME TAX. Schedule C, D and E, Income, is. in the . EXEMPTION AND ABATEMENTS. Income not exceeding 160 Exempt. Excd. Not Excd. . 1 60 400 Abatement of 160 400 500 150 500 600 i 20 600 700 70 Gross Income. Income Tax. Net Income. w ^450 ? p (*) ? 20 p w p 31- ios. p (d) ^1500 p p (e) ? ? /76o (/) ? p ^340. IQS. 254 A MODERN ARITHMETIC 23. Estate Duty is charged at the following rates : Duty. ESTATE. per cent. * zoo and does not exceed 500 - - i . o 500 1,000 - -2.0 1,000 10,000 - -3.0 10,000 25,000 -4.0 25,000 50,000 - - 4 . 10 50,000 75, - - 5 75,000 100,000 - - 5 . 10 100,000 150,000 - -6.0 150,000 250,000 - - 6 . 10 250,000 500,000 - -7.0 500,000 1,000,000 - - 7 . 10 1,000,000 - 8 . o Find the Estate Duty paid on the following (true to the nearest penny in all cases) : (a) 689. I2S. (b) ,18526. (c) 3637- (d) .189,764. (*) 689,124. (/) ^532. (g) 1896. (h) 7632. ios. (t) 600,807. O) 178,981. (*) 631. (/) 2563. (*) 7342. () 3,189,763. 24. Calculate the original value of an estate if, after paying Estate Duty, its value was as below : () 396. (b) 6596. ( f ) 5432. (d) 73,7io. (*) 4850. (/) 7881. 5 j. (g) 16,320. (//) ,518,000. (Duty at the rates given in Ex. 23.) 44. ERRORS IN DATA. In numerous cases the data, that is, the given or observed facts or quantities available in a calculation, are not quite accurate; for example, a man wishes to calculate the area of a rectangular plate of metal ; he has his foot rule for ERRORS IN DATA 255 measuring the length and the breadth, and with this he finds, true to the nearest tenth of an inch, that the plate is, say, 5-4 inches long. 4-5 broad. The calculated area is therefore 5-4x4-5 sq. inches. Multiply 5-4 4-5 21-6 2-70 24-3 The area works out to be 24-30 square inches. But when the length is taken as 5-4 in., all that is meant is that the length is nearer 5-4 in. than either 5-3 or 5-5, and similarly that the breadth is nearer 4-5 in. than either 4-4 in. or 4-6 in. The length might actually be either nearly 5-45 in. or only slightly over 5-35 in., while the breadth might actually be either nearly 4-55 in. or only slightly over 4-45 in. The difference between the 5 8 c' FIG. 36. greatest and lowest value for the area would, in the accompany- ing figure, be represented by the shaded figure BDCC'D'B', which clearly has the area of (5*35 x 0>I ) + (4'45 x o-i) + o-oi, i.e. 9-8 x o-i +0-01 = 0-98 + 0-01 =0-99, or practically one sq. inch. The estimated value might be therefore 0-5 sq. in. too large or too small. 256 A MODERN ARITHMETIC EXERCISES LXV.a. Graphic. \Apparatus : Squared Paper. Mathematical Instruments^ 1. A rectangle is measured by a ruler reading only to 0-2 of an inch ; its length is taken to be 3-6" and its breadth 2-4". Draw the rectangle according to these measurements ; show also the smallest rectangle and the largest rectangle consistent with 3-6" and 2-4" as the nearest readings true to 0-2" ; find the difference in the areas and the possible error in taking the area as being 3-6" x 2-4". Verify your work by calculation. 2. With an ordinary foot rule, divided into inches and eighths of an inch, the side of a square is measured and taken to be 4^" ; find graphically the possible error in taking (4^) 2 sq. inches as the true value of the area. 3. A rectangle is measured by an ordinary millimetre scale ; its length is estimated at 12-6 cm., its breadth 6-4 cm. What error might there be in the area as calculated from these data ? 4. A circular disc is measured by a foot rule, graduated to tenths of an ' inch ; its radius is taken to be 3-2 inches. Find graphically the possible error in working out the area from this value of the radius. EXERCISES LXV.b. Note. In the following exercises it is assumed that there is no knowledge of the figures beyond those given, but that the figures cited are the most correct to the given degree of accuracy. Thus, 3-7" means something greater than 3-65", and less than 3-75", i.e. 3-7" is nearer to the correct result than either 3-6" or 3-8". Hence, 3-7" would not mean exactly the same thing as 3-70", 3-70" meaning a length less than 3-705", greater than 3-695", i.e. 3-70" is nearer the true length than either 3-71" or 3-69". LIMITING VALUES 257 1. The sides of a rectangle are taken to be 6-8" and 5-6" ; find the difference between the least and greatest possible value of the area. 2. A rectangular block is measured, and the following values entered : Length 12-2", Breadth 8-7", Thickness 6-0". What is the greatest possible volume of the block, and what its least volume ? 3. The contents of a rectangular cistern are calculated from measurements of its dimensions ; these are taken as 100-2 cm. 50-4 cm. 40-6 cm. What error may exist when its contents are expressed in litres ? 4. The dimensions of a rectangular grass plot are measured by a chain, and are found to be 10 ch. 12 links ^ 8 h 8 1' k I ( nese can only be true to the nearest link.) Find the difference between the least and greatest possible area as deduced from the measurements. 5. The sides of the following squares are measured with a plain scale graduated to tenths of an inch ; find the least and greatest values of the areas : (a} 3 ft. 4-5 inches, (b) 11-2 inches, (c) I ft. 6-8 inches. (d) 2 ft. o-i inch. (e) 25-3 inches. (/) 5 ft. o inches. 6. The length of a rectangular block is measured and taken as 7-9 cm., its breadth 4-6 cm., thickness 3-4 cm. and specific gravity is estimated at 8-8. Find the limits to the possible weight of the block in grams. (By specific gravity is meant the number obtained by dividing the weight of any volume of the substance by the weight of an equal volume of water.) J.M.A. R 258 A MODERN ARITHMETIC 7. Evaluate in the same way the limits to the possible weights in grams of the following blocks, stating the answer to the units place only : Length. Breadth. Thickness. Specific Gravity. (a} 30-2... cm. 10-3... cm. 5-0... cm. 7-2... cm. (V) 18-5 12-1 4-2 7-2 (c] 20-5 10-5 6-1 1 1 -2 (d} 16-1 16-1 16-1 1 1-2 (<?) 13-2 13-2 13-2 0-988 (/) 60- 1 40-4 -2o-g 0-643 8. A sheet of metal is weighed and tound to be 33-24... grams , its length is measured and taken as 20-2... cm. and breadth 10-1.. cm. ; its specific gravity is taken as 7-2.... Find the limits to its thickness in millimetres. 9. Find the limits true to o-oi cm. to the values of the thickness in the following blocks : Volume. Length. Breadth. (a) 763-5 c.c. 1 8- 1 cm. 7-1 cm. () 1000-3 c - c - 20-0 cm. 10-0 cm. (c) 1976-2 c.c. 35-9 cm. 8-3 cm. (d) 968-0 c.c. 29-5 cm. 17-5 cm. (<?) 64-5 c.c. 8-1 cm. 3-2 cm. (/) n8-6c.c. 10-2 cm. 4-6 cm. 10. In order to calculate the length of wire in a coil, the follow- ing operations were performed : (a) The coil was weighed. (b~) A piece of wire was cut off and its length determined. (c) The piece of wire was weighed. If the weight of the coil was 976-35... grams, the piece of wire 100-5... cm - !ong and 1-03 grams in weight, find the limits to the length of the wire in the coil in cm. 11. At a school sports, the time for the winner of the 100 yds. race was given as ill sees. If this time was taken by a stop- watch reading to the nearest fifth of a second, find the limits to the speed of the boy in miles per hour. (Neglect errors in the length of the course, the personal errors of the time-keeper, etc.) LIMITING VALUES 259 EXERCISES LXV. c. Practical. [Apparatus : Blocks. Balance. Plates of Metal. Wire.} [When it is not convenient for the following operations to be performed by the class, they may be made by the teacher, and the results given to the class.] 1. Find the specific gravity of the given block from measure- ments of its weight, length, breadth and thickness. Show clearly the limits to the possible value of the specific gravity. 2. Find the weight of the given piece of metal in and out of water ; find the loss of weight when weighed in water, also the ratio of the weight of metal to the loss of weight in water ; indicate in your answer those figures the accuracy of which you can rely upon. 3. Find the volume of the given rectangular plate of metal by (a) finding the volume of water it displaces ; (b) measuring its length, breadth and thickness ; find the limits to the values according to the two methods, and point out which is the more accurate. 4. Measure the volume of the given wire by (a) finding the loss of weight in water ; (b} finding its length and cross section. Estimate whether the difference in the values found by the two methods is greater or less than it ought to be. 45. APPROXIMATE OR CONTRACTED MULTIPLICATION. It has been shown that not only is it unnecessary to carry calculations beyond a certain point, but that generally the data supplied are not absolutely correct. Special methods 260 A MODERN ARITHMETIC are therefore usually employed for shortening processes of calculation. One of the most general and useful is that of contracted multiplication. EXAMPLE. Find the volume of a block the length of which is 18-23", breadth 17-16", thickness 14-12" (each true to o-oi"). Proceed by multiplying out the product of 18-23, 1 7 >l6 > 14-12. 18-23 17-16 3128-268 1251-3072 31-28268 6-256536 4417-114416 The volume would therefore be 4417-114416 cubic inches. Now, not only would it probably be unnecessary to know the volume to its millionth part, but a large number of the figures in the multiplication are absolutely wrong. For the length is not really 18-23 i n - exactly; there must be figures after the 3. Call the ist of such figures a. Similarly in the breadth there probably is an unknown digit b after the 6 ; and an unknown digit c in the expression 14-12 for the thickness. Go through the multiplication again, marking those figures which are probably wrong with a light line drawn through them, and putting an asterisk for a figure (due to a, b, c, etc.), about which we know nothing. APPROXIMATE OR CONTRACTED MULTIPLICATION 261 18-230: 182-30: . .. . i27-6x*. . . 1.8230 . . 1-093^*- ***** 312- The i in the second decimal place is probably wrong, as when digit a is multiplied by 7 a figure would have to be carried over. For a similar reason this 8 is wrong. Figures exist here on account of digit b. Note also that an error might exist in the 8 in 3I28-XW* 14-12*: the first place of decimals. 3I-2 6-256$^** ********* Clearly no reliance can be placed on any of the decimal figures, while the 7 in the units place is not to be relied upon as absolutely correct. The work might therefore have been shortened by omitting figures for which no authority exists. Thus: i8-X$ 17-16 182-3 127-61 Here only 18-2 is multiplied. 1-82 18 1-09 312. 14-12 Multiply only 312-8. >, 312. 3128- 1251-3 3i-3 6-3 No figures are set down beyond the 2nd decimal place ; the 2nd decimal place result, 2, is not correct ; neglect it therefore in evalu- ating the next product. 44 I( H This 9 is wrong, but being certainly greater than 5 the result is taken as 4417. 262 A MODERN ARITHMETIC 5^-4374 Even if the given figures to be multiplied are absolutely correct, the product may only be required to a certain degree of accuracy. Thus : To find the product of 313-4567 and 518-4374 true to 2 decimal places. Figures in the 3rd place of decimals will certainly affect the result in the 2nd place. Figures in the 4th place may affect those in the 2nd. Now strike out the 7 in the multiplicand, but since 4x7, i.e. 28, is nearer 30 than 20, carry a 3. Now strike out the 6 in the multiplicand, but carry a 2, since 3 x 6 is nearer 20 than 10. Now strike out the 5 in the multiplicand, but carry on a 4, since 7 x 5 + ... is nearer 40 than 30. 2507-6536 125-3827 9-4037 2-1942 _ -1254 162507-67^ The result is 162507-67, where 7 is the actual figure; or 162507-68, where 8 is the nearest figure in the 2nd decimal place. More frequently it is required to evaluate results true to a certain number of significant figures, e.g. if the result reaches to millions, units may be of small significance, whereas if the result is very small millionths may be important. APPROXIMATE OR CONTRACTED MULTIPLICATION 263 EXAMPLE. Evaluate 876208x31-6243 true to ^significant figures. In most of these cases it is convenient to alter the form of the number as follows : 876208 = 8-76208 x io 5 . _ 31-6243 = 3- 16243 x Jo 1 - ~ The product is 8-76208 x 3-16243 x io 6 , and if we work true to the 4th decimal figure our result will be certainly correct. 3-16243 26-28624 -87621 52572 1752 35 26 27-709^ The product is therefore 27-70945 x io 6 , or, true to the nearest 4th significant figure, 27710000. In most cases, and especially where the calculations are long, it is desirable to test roughly whether the answer found is a likely one. Such tests should be mental. Thus 876208x31-6243 = 8-76208 x 3-16243 x io 6 . 8-76208 is slightly over 8|. Now, 8x3-16 is slightly over 25. f of 3-16 (nearly f of 3-2) is about 2-4. The answer is near, therefore, to (25 + 2-4), i.e. 27-4 millions. Or, it might have been reasoned that 0-16245 is nearly \. 3 times 8-762 is 26-3... . J 8-76 1-4.... The answer, therefore, is 27-7 millions 'about.' 264 A MODERN ARITHMETIC EXERCISES LXVI. a. Mental or Oral. Evaluate the following to the degree of accuracy stated 1. 981 X99 to the nearest 1000. 2. 326x650 to 2 significant figures. 3. 801 X420 to 3 4. 4-26 x io 2 x 2 x 3-141 to 2 significant figures. 5. 958x167 to 2 significant figures. 6. 1836x2482 7. 0-842x4976 8. 693-2x3358 9. 0-0628x0-07018 10. 1764x996 11. 4018x7492 12. 2497x0-08653 13. 38600x0-02496 14. 36-18x110896 15. 248x4-1247 16. 7496x3987 17. 934x101 18. 696x5-167 19. 333X3997 20. 876x0-0002004 ,, 21. 119x0-374 22. 128200x0-8748 ,, 23. 3624x0-00498 24. 998x6-049 ,, 25. 397X7-332 APPROXIMATE OR CONTRACTED MULTIPLICATION 265 EXERCISES LXVI. b. NOTE. The figures given are approximate only. 1. Find the weight of a block of aluminium bronze 5643-2... cubic centimetres in volume, the weight of each cubic centimetre being 8-372... grams. 2. The following is an extract from a table of births and deaths for the year 1903 ; find, as accurately as you think the data allow (i.e. to the nearest unit, or ten, etc.), the actual number of births and deaths in each town : Cities, Boroughs, and Urban Districts. Population. Ratio per 1000. Births. Deaths. () London (the Metropolis) - 4,6l3,8l2 28- 4 15-7 (*) Aston Manor - 79,417 28-7 13-9 w Birkenhead ... H3,598 30-8 16-8 faO Birmingham - 533,039 3i-8 17-8 w Blackburn 131,218 25-1 i5-7 (/) Bolton .... 173,401 27-0 17-5 w Bootle .... 60,761 33-o 19-0 (X) Bournemouth - 63,132 17-8 I 2- 1 Bradford (Yorks) 283,412 23-3 l6-4 ( Brighton - 125,405 24-3 14-3 Bristol .... 338,895 27-4 14-3 (1} Burnley .... 99,469 27-2 19-2 (m) Cardiff .... 172,598 30-5 I4-0 3. The chemical composition of dry air is as follows : By Volume. By Weight. Nitrogen - - 78-40 per cent. 75'95 per cent. Oxygen- - - 20-94 Argon - - - 0-63 Carbon Dioxide - 0-03 23-10 0-90 0-05 Find the volume of nitrogen in a room 6-32... metres long, 5-46... metres broad and 3-27... metres high. State your answer true to the nearest cubic metre. 266 A MODERN ARITHMETIC 4. The length of a rail of iron is 24-321 ... ft. on a cold day ; what will be its length on a day when each foot of the iron has expanded to a length of 1-00324... ft. ? 5. What will be the yield in ounces of gold from 16,800 tons 4 cwt. (to the nearest cwt.) of ore, if each ton contains 3-426... oz. of gold ? 6. The chemical changes which have taken place during the conversion of vegetable matter into the several varieties of coal may be seen by a comparison of their percentage analyses, as given in the following table : Wood (Oak) - Peat (Irish) .... Lignite (Bovey Tracey) - Bituminous Coal (Newcastle), - Steam Coal (South Wales) Anthracite (South Wales) - Find the weight of carbon in the Bituminous coal brought up from a seam 5 acres in area and of average thickness 59-1... in., assuming that each cubic yard of coal in the mine yields I ton at the pit mouth. 7. The height of the barometer decreases as the elevation above the sea-level increases. Change the following table into one where the height of the barometer is in inches and the elevation in feet (assume that i metre = 39-37 in.) : Carbon. Hydro- gen. Oxygen and Nitrogen. 48-94 5-94 45-12 55-62 6-88 37-5 69-94 5-95 24-11 88-42 5-61 5-97 92-IO 5-28 2-62 94-05 3-38 2-57 Elevation in metres \ above sea-level. J 100 200 300 400 500 Height of barometer! in millimetres / 760 751 742 733 724 716 8. What is the number of cubic metres of air in a room 6-426... metres long, 5-821... metres broad, 3-872... metres high? (State answer to the nearest trustworthy figure.) APPROXIMATE OR CONTRACTED MULTIPLICATION 267 9. The length of the stroke of the piston of a steam engine is i6"-oo and the number of strokes made per minute is 148. Find the actual travel of the piston per hour, expressed to the nearest mile. 10. Find to the nearest 1000, the number of bricks in a rect- angular kiln the internal dimensions of which are 23 ft. long, 14 ft. wide and u ft. high, assuming each brick to occupy 100 c. in. and deducting a space of 0-6 per cent, of the capacity for setting the brick. Evaluate the following to the degree of accuracy stated. (In the following examples the figures given are supposed to be exact.) 11. (625-024) 2 true to units' place. 12. 872-1256x375-2164 i st decimal place. 13. 963.34x64-234 ist 14. (172- 131)2 nearest unit. 15. { 4- 1 8765 } 2 4th decimal place. 16. 89-006x91-083x103-025 ist 17. 2-867x3.424x1-1 2nd 18. 103-6x102-1x92.8 units' place. 19. (i-02) 3 4th decimal 20. (2-08 1) 3 2nd 21. 892-562x2-018643 true to 5 significant figureSo 22. 102-345x97-655 5 23. 324-613x133-475 ,", 4 24. 791-82x36325 5 25. 36-2345x99875 4 26. 897699x19-9891 6 27. 7789-13x234-56 4 A 28. 3456-2 x 1903 5 29: 3876 x 10401 3 30. 9969x8858 4 268 A MODERN ARITHMETIC 46. CONTRACTED DIVISION. For reasons similar to those described in the case of multiplication, contracted methods are often employed in division. EXAMPLE A. To find the weight per foot of a cable when it is known that a piece estimated as 235-24... //. long weighs 928-34... Ibs. The fraction ^ 2 '34--- mus t be evaluated, and in all 235-24... probability figures should follow the 4 in the numerator and also the 4 in the denominator. Suppose the figures following are ab... in the numerator, a/3... in the denominator. 928-34^... _ 9-2834^... 235-24(1/3... ~ 3-946 Here a line is drawn through every which is not correct 2834^ 7 . ot - 7 V I The 2 may not be correct, since a figure I may be carried over from the a x 3. / The fig ure to be brought down is probably \ not a -zero ; it is represented as a. 2-1171)^ I4H44 ^^ Without a single correct figure. CONTRACTED DIVISION 269 So far as the quotient is concerned, notice that the correct figures are the same as if the following process was adopted. 3.946 2-352 9-2834 7-0572 2-2262 Instead of bringing down a zero, cut off the last figure in the divisor, but carry on 2-1172 a 4 in multiplying by 9. 2-35 1090 So cut off the last figure 2 in the divisor, 941 carry a i, since the 4x2 is nearer 10 than o. 2-3 149 Similarly cut off the 5. ill It is not necessary to set down the divisor each time ; the last digits might merely be crossed out lightly, while the procedure of the Italian method might also be adopted. The example appears thus : 3-946 = Quotient. 2-35XXI 9-2834 2-2262 1090 49 8 A similar process is used when the divisor and dividend are supposed exact, but where the result is required true to a certain number of significant figures, or to a certain number of decimal places. EXAMPLE B. Evaluate 89356432-^-234643 true to 4 signifi- cant figures. 8935 6 432 -5- 234643 = 893'5 6 432 -5- 2-34643- Arrange as in Short Division, 270 A MODERN ARITHMETIC 2-. ..[893.56432 abc-de The digit (a) of highest place value in the answer is hundreds digit ; the fourth (d) is in the first place of decimals, and the fifth (e) in the second place of decimals. This digit (e) is determined by the fifth significant figure (6) in the dividend, and the figures 432... beyond this might have been regarded as unknown, so we follow the method used in Ex. A. (To know the result true to the 4th significant figure (</), it must be known whether the 5th significant figure (e) is greater or less than 5.) EXAMPLE c. Evaluate 29356432-7-834643 true to 4 signifi- cant figures. Arrange as in B, 8... 293-56432 ab-cde It is clear that six significant figures must be written down in the dividend. It will be safer to proceed thus in all cases : Write down the dividend to two significant figures beyond the number of significant figures required in the answer, and proceed as in Ex. A. The work in B would therefore appear as below. Thus 380-8 = Quotient. 2 -346 4 X | 893-56*4 2-346 189-63 2-34 1-92 2-3 1-92 2 4 Here, and always in future, make a rough approximation to the answer first. Thus : 89356432-^234643 = 893-5... -T- 2-34..., and the quotient clearly lies between 446 and 297 (taking as divisors 2 and 3), and it is nearly the same as 893 -7- 2^ = about \ of 2700 ' =380. Checks of this kind are only rough, and should be mental. CONTRACTED DIVISION 271 Consider another rough mental calculation. 345.632x24.365-7-143-89 3.456 x io 2 x 2-436 x 10^(1-43 x io 2 ) = 10 X 3-456 X 2-436 -M -43. For a rough guess at the result, replace 3-456 by 3-6 2-436 by 2-4 1-43 by 1-44 The result will not differ much from IOX3 ' 6x2 ' 4 1-44 i.e. 345-632 x 24-365 4- 143-89 = 60 approx. If the answer be found very much smaller or very much larger than this, an error has probably taken place. Later on, methods will be given for approximating still more accurately. EXERCISES LXVII. 1. What is the specific gravity of a substance the weight of which in air is 819-324... grams and weight in water is 692-456... grams ? 2. Find the number of gallons of water in a rectangular cistern measuring 12 ft. 5 in. long by 8 ft. 4 in. broad and 6ft. o in. deep, assuming 277-264... c. in., to the imperial gallon. (State your answer to the nearest unit.) 3. What weight of steam will be used per minute in filling a cylinder the capacity of which is 1924 cubic inches, if the piston makes 50-0 strokes per minute, and if 4-33 cub. ft. of the steam weighs i Ib. ? 4. If a metre be 39-37011... inches, convert (a) 2394-622 inches into metres, etc. (b) 8 metre, 2 dm. 3 cm. 4-2... mm. into ft. and inches. 272 A MODERN ARITHMETIC 5. Find the ratio of the weight of sediment carried to the amount of carrying water in the following rivers : Discharge in thousands of cubic ft. per sec. Annual Discharge of Sediment in tons. Ratio. Danube Nile Rhine 315 65-8 . 108 million 54 million 36 million I : i : i : State your answer in the form i : x, where x is the weight of water for every unit of sediment carried. (Assume the weight of i cub. ft. of water to be 62-5 Ib.) 6. Find in how many years the drainage area of the following rivers would be lowered by an average amount of i foot, on account of the action of the water : (i) (2) Area of Basin in Square Miles. Annual Discharge of Sedi- ment in cubic feet. (a) Mississippi - 1,147,000 7,459,000,000 Danube 234,000 1,253,739,000 M Upper Ganges - 143,000 6,368,077,000 (d) Po - 30,000 . 1,510,137,000 (A Rhone 25,000 600,382,000 7. Find the circumference of a circle the radius of which is 89-324 ft. (^ = 3-14159....) 8. Find the circumferences of the circles the diameters of which are (a) 318-12... (b) 9-9521... (c) 8202-1... (d) 0-26002... (*) 66211 (/) 33-914. 9. Find, as correctly as your data will allow, the radius of a wheel the circumference of which is 48-256 inches. (^ = 3-14159... .) 10. If a Norwegian mile be 11-299 kilometres or 7-021... English miles, express an English mile in kilometres. SQUARE ROOT 273 47. SQUARE ROOT. When 12 is multiplied by 12 the product = 144, or i2 2 = 144. 144 is the square of 12, and 12 is called the square root of 144; represented symbolically, 1 2 = ^144. Similarly, 3 = \/9> o-i =/s/o-oi, \ = I J\- Extracting square roots by factorizing. EXAMPLE. 71? obtain the square root 0/" 435600. Split up 435600 into its prime factors; 3 2 x 2 4 x 5 2 x ii 2 is obtained. .'. 435600^(3 x 2 2 x 5 x n)(3 x 2 2 x 5 x n) .*. \/4356oo = 3X2 2 x5xn = 660. EXERCISES LXVIII. a. Mental or Oral. 1. State the numbers of which the following are the square roots: i, 2,..., 12, 13, 14, ...20. 2. What must be the last digit of any number having an exact square root ? EXERCISES LXVIII. b. Obtain, by factorizing, the square roots of 1. 1764. 2. 44100. 3. 5625. 4. 53361. 5. 7056. 6. 1296. 7. 2304. 8. 2371600. 9. 2916. 10. 3969. 11. 46656. 12. 4356. 13. 65536. 14. 1334025. 15. 36864. EXERCISES LXVIII. c. Mental or Oral. 1. How many digits will there be in the squares of (a) 8, '() 13, ( c ) 386, (d) 89765, (e) 13692, (/) 4789862? J.M.A. S 274 A MODERN ARITHMETIC 2. How many decimal places in the squares of (a) 0-8, (b) 0-08, (V) 0.000636, (</) 0-2062, (*) 0-5683, (/) 0-00073? 3. How many digits will there be in the integral and decimal portion of the squares of (a) 2-6, (b) 3-006, (c) 563-0082, (d) 763-0063, (e) 23-86745 ? 4. How many digits will there be in the square roots of () 64, (J) 121, (*) 169, (d) 289, (') 9, (/)3969, () 46656, (A) 44100? 5. How many decimal places will there be in the square roots of the following (each of which has an exact square root) : (a) 0-64, (b) 0-0064, (c) 0-0324, (d) 0-046656 ? 6. State the number of figures in the integral and in the decimal portions of the square roots of () 39-69, W 4-6656, (c) 29-8116, (d) 533-61, (e) 530-8416. General Method. Graphic Illustration. Suppose the square ABCD represents the number, and that it is known that kbcd, representing say n 2 , B C is very nearly equal to it, so that n is a sort of first guess to the square root of the number. The problem is therefore to find the additional amount B, representing say x, which has to be added to n to obtain the required square root. From the figure it is clear that what is wanted is a number x t such that twice the product of x and , together with the square of*, is equal to the difference between the square of n and the given number. Further, that ABCD-Afo/ is very nearly 2 x x, and therefore that x is suggested by (ABCD - Meet) + (2*1). This is the key of the Arithmetical Method of extracting square roots. SQUARE ROOT EXAMPLE. To evaluate ^4225. 275 Beginning with units and tens digits, divide up the number so far as possible into ' periods ' of two digits. 5 = Units 1 portion of ^4225 ; 60 Twice 60 = 120 Add the 5 5 125 42(25 3600 6 25 625 6o = Tens/Y 4225 lies between 6o 2 and yo 2 . Set down 60 on both sides ; multiply them as shown. ... =4225 - 6o 2 ; 4225 - (6o) 2 -^ 120 sug- gests 5 as units digit. Multiply 125 by 5. Since there is no remainder 65 =^4225. EXAMPLE. To evaluate ^104976. Divide into periods as in the preceding example, beginning with units and tens digits. 4 = Units ] portion of 20 = Tens ^^104975. 300 Twice 300 = 600 Add 20 20 620 Twice 320 = 640 Add 4 4 644 >|49l?6 90000 4976 i 2400 2576 2576 300 = Hundreds] y 104976 lies between 3oo 2 and 4oo 2 . Set down on right and left side as shown ; multiply. ... = 104976 - 3oo 2 ; division by 2 x 300 suggests 20 as tens portion of J. Multiply 620 by 20. ... = 104976- 32o 2 ; division by 640 suggests 4 as units figure. Multiply 644 by 4. Since there is no remainder ^104976 = 324. 276 A MODERN ARITHMETIC As in division and multiplication, when the processes are clear, the cyphers are omitted, and the work abbreviates into 3 2 4 = 324 149 124 Or: 3 62 644 644 2576 2576 where, on the right, methods similar to the Italian methods in Division have been adopted. There is no difference of procedure when finding the square root of a number partly integral and partly decimal. The following is an example worked out fully : Dividing into periods on both sides of decimal pt. 7000 |6i|32|8i|63|-3i|29 1 49 oo oo oo i2|32|8i|63|. 3 i|2 9 14800 j_i840ooo twice 7800= 15600 j a ~ 15630 48|8i|6 3 |. 3 i|92 468900 twice 7830=15660!^ 15661 twice 7831 = 15662 JI92I63I-3II92 i 5661 add 36|02|. 3 I|2 9 3I32-44 =7831 03, 7831-23 twice 15662-2 7831-2 = 15662 -4^ add 15662-43 And Abbreviated, the procedure would be as follows : 7831-23 7|6i|32|8i|63|. 3 i|2 9 | 148 1232 1563 4881 15661 19263 15662-2 3602-31 15662-43 469-8729 SQUARE ROOT 277 EXERCISES LXVIIL d. 1. Find the sides of the squares the areas of which are : (a) 169 sq. ft. (H) 289 sq. cm. (c) 529 sq. yards. (d) 841 sq. cm. (e) 2209 sq. in. (/) 2809 sq. ft. 2. What numbers when multiplied by themselves yield (a) 3364. () 4096. (c) 5476. (d) 7225. (/) 7921. (/) 8649. (g) 9409. (//) 9801. 3. What length of fencing will be required to go round the following square plots : (a) 15876 sq. yds. (b) 64009 sq. yds. (c] 94249 sq. metres. (d) 191844 sq. ft. (e) 358801 sq. ft. ? 4. Evaluate (6) ^502681. (e) \/4i6568i. (a) ^451584. (d) 7893025. (o) A/I 6 1 44324! (k) ^281 53636. (c) (/) V 1 4992384. (z) \/2 5 847056. (0 0-853776. CO 905-43819025. (z) 0-000079281216. 0) ^5035747369. 5. Find the square root of (a) 530-8416. (b) 4911-486724. (d) 241690-2244. (e) 43-1649- (g) 0-00279841. (h) 0-0006441444 (/) 4096-896049. 6. A company of soldiers 864 strong is to be formed into the largest possible solid square. What is the number in the side of the square ; how many men if any are left over ? 7. What length of wire netting will be required to enclose a square lawn of i acre area ? (Answer to the nearest yd.) 8. From a rectangular piece of glass 19-4 cm. long and 14-8 cm. broad, the largest square piece is cut off. What is the area of the remaining rectangle, and what of the square ? 9. Three square flower beds, of 4 ft. 6 in., 5 ft. 9 in., and 7 ft 9 in. side are to be replaced by a large square one equal in area to the sum of the three. What must be the side of the square (true to the nearest inch) ? 278 A MODERN ARITHMETIC 10. A rectangular lawn measuring 196 ft. by 169 ft. is to be replaced by a square one equal in area ; what is the length of the side of the latter ? Compare the lengths of fencing necessary to enclose each. 11. The expense of a party during an outing was i. los. \d., and it was found that the expense in pence per head was equal to the number forming the party. How many formed the party, and what was the expense per head ? Square Roots of Vulgar Fractions. The following methods may be adopted in different cases : (1) Extract the square roots of numerator and deno- minator separately. (2) Change into an equivalent fraction by multiplying numerator and denominator by the denominator (this may frequently be shortened). (3) Change into a decimal and extract its square root. Method (i) is of use when both numerator and denominator have exact roots. EXERCISES LXVIII. o. Evaluate the following : (o ^m_ (2) (4) J3Mj> (5) (?) ^Bi to two decimal places. (8) v T 7 /g to two decimal places. (9) ^H to two decimal places. 48. INTEREST. When a sum of money is lent to a Bank (or placed on Deposit) or to a borrower, the money paid for the use of such sum is known as Interest ; the money lent is frequently termed the Principal. INTEREST 279 In the Post Office Savings Bank, the regulations as to Interest are governed by the following rule: "X. Interest at the rate of 2. los. per cent, per annum is allowed on every complete pound deposited, and commences on the first day of the month next following the deposit, and is calculated to the 3ist of December, when such interest is added to the principal and begins to carry interest as an ordinary deposit." EXAMPLE. A person puts sums of money into the Post Office Savings Sank, as indicated by the following table: Date of Defosit or of Warrant, etc. Amount of Deposit in words or Number of Warrant in figures, etc. Amount of Deposit. IQOI May jo August 4 October 17 Two pounds, ten shillings Three pounds^ twelve shillings Seven pounds, fifteen shillings . 3. d. 2 IO O 3 12 7 15 o What does his interest and principal amount to at the end of the year ? Through June, July and August (3 months) the money bearing interest is 2 [the odd shillings do not bear interest, while the 3. I2S. od. will not begin to bear interest until September i]. On ;ioo the interest for 12 months is 2. los. od. .'. 100 3 months is o. 125. 6d. '. 2 ,, 3 months is -^ (o. 1 2s. 6d. }i.e. $d. After September i, the person has 6. 25. od. in the Bank, 6 of which bears interest, and until November i (2 months), he therefore receives interest on the 6, and the amount of this interest is determined as before, .e. IOO or 280 A MODERN ARITHMETIC From November i to the end of the year, the deposit bearing interest is 13- The interest on this for 2 months is TZHT of TZ of > 2 - IOS - ^> *'' innr of 2 - IOS - d - or Z 3^ The total interest is therefore [3^. + 6^. + 13^.], i.e. is. icd. The amount to the person's credit at the end of the year is therefore 2. IQS. od.+$. i2s. 0^.4-^7. 15.?. od. + o. is. iod., i.e. ^13. 18^. lod. Here, and in many similar cases, the interest can be calculated more quickly as follows : The interest on i for i year is -^ of i or 6d.; the 100 interest for i month is therefore \d. Therefore the interest is (2x3 + 6x2 + 13x2) half- pennies or o. is. lod. Similarly, if the rate of. interest had been 5% per annum, each j would bear id. interest per month. EXERCISES LXIX. a. Find the amounts due to depositors in the Post Office Savings Bank on Dec. 3ist ; in the following cases : Date of Deposit or of Warrant, etc. Amount of Deposit in words or Number of Warrants in figures, etc. Amount of Deposit. 1903 s-d. Jan. 10 Eight guineas - ... 880 Mar. 17 Eight guineas .... 880 May 24 Eight guineas .... 880 July 31 Ten guineas - ... 10 10 Sept. 7 Ten guineas .... IO IO O INTEREST 281 2. Date of Deposit or of Warrant, etc. Amount of Deposit. 1902 s. d. Feb. 3 5 12 o Mar. 10 4 17 o June 16 7 10 o Nov. 22 990 3. Date of Amount Deposit or of Warrant, etc. of Deposit. 1903 * d. Feb. 17 9 ii o Sept. 26 7 16 o Nov. 13 8 18 o Dec. 10 5 H o EXAMPLE. The books of the depositors below were not forwarded to be made up at the regular time. Find the amounts due to the depositors at the ends of the last years in which entries are quoted. Date of Deposit or of Warrant, etc. Amount of Deposit in Words or Number of Warrant in figures, etc. Amount of Deposit. KJ02 s.d. February 15 Ten pounds only 10 October 10 Eight pounds, eighteen shillings 8 18 o 1903 January J Eight pounds only 800 May 6 Six pounds, fifteen shillings 6 15 o July 16 Nine pounds, twelve shillings 9 12 o In 1902 the depositor has IQ bearing interest for 10 months and ^8 for 2 months. The interest is, therefore, (100+16) halfpennies, or 58^., i.e. 4s. lod. The amount to his credit on Jan. ist, 1903, is ;i8. iSs. + 4S. iodT., i.e. 19. 2s. io</. Of this sum ,19 bears interest. He therefore has as interest in 1903 [19 xi2 + 8xu+6x7 + iox5] halfpennies = 12.$*. At the end of the year the amount which serves as principal for 1904 is ig. 25. io</.+;8 + ;6. 15^ + ^9. I2S. + 12S., i.e. ^44. is. -Lod. 282 A MODERN ARITHMETIC EXERCISES LXIX. b. Find, as in the worked example on p. 281, the amounts due to the depositors at the end of the last years in which deposits are entered. 1. 2. 3. Date of Deposit or of Warrant, etc. Amount of Deposit. Date of Deposit or of Warrant, etc. Amount of Deposit. 1902 *.d 1901 s.d. Mar. 1 8 IO IO O Sept. 20 12 1903 1902 Aug. 1 6 8 10 o Aug. 10 14 o o Nov. 1 6 700 Nov. 5 12 Date of Deposit or of Warrant, etc. Amount of Deposit. I9Oi June 3 J- d. 10 1903 June 2 July 3 20 IO O O EXERCISES LXIX. c. Mental or Oral. What interest will accrue to 1. ^50 in 12 months at 2^ % payable yearly? " As* 5> I J )5 jj 3. .40 i 4. .50 3 5. ^80 2 6- ^80 i On what sum will the interest amount to % payable yearly ? 7. i. in i year at 2^ 8. \d. i mo. 10. ios.6i 6 11. 125. 55 8 12. ISA 5J 3 )> )) SIMPLE INTEREST 283 In how many months will the interest on 13. i t at 2^ % payable quarterly, amount to \d. ? 14. 10, y.^d. 15. 40, 105. 16. 7, y. 6d. 17. ;i8, y. 18. 72, i. 7-f- Simple Interest. In most cases in which money is deposited in banks, the interest is added to the principal at fixed intervals, the sum obtained then serving as the new principal. Frequently other arrangements may be made, in which the interest-earning money remains unchanged ; the interest charged or earned in such cases is known as Simple Interest. The following worked examples will illustrate the method of procedure : EXAMPLE. A person, A, lends another, B, the sum of ^350 for 4j years, at 2\ per cent, per annum Simple Interest. What interest does A receive during that time ? This statement means that for every ;ioo lent to B, A receives 2^ yearly as interest. The interest, therefore, received by A is f f x 2 \ every year. The interest received by A in 4^ years is 4^ x J x 2\ = 39' is. 6d. EXAMPLE. What is the simple interest received on 1 75-. 6d. in 2 years 3 months at 3 % per annum. State the answer true to the nearest farthing. Interest 5 T 2-87 5 ^512. lyj. 6d. x T ^ in i year _ 3 ^512-875 x T ^xf in 2 yrs, 3 mo. 1538-625 400 j 13847-625 34-6i9 284 A MODERN ARITHMETIC EXAMPLE. At what rate per cent, per annum will the simple interest on ,475 become 66. IQS. in 4 year -s ? This is equivalent to asking what the interest would be on ;ioo in i year under the above conditions. Interest on ^475 in 4 years = ^"66-5. i year = J of ^66-5, and ;io in i year =^| of J of 66-5 Hence the rate per cent, per annum was 3^. EXAMPLE. In what time would a man have received 2$. ios. od. as interest, if he lent another 126. 55-. od. at 3 % P er annum Simple Interest ? The interest on ^126. 55. od. in i year at 3 % is T of ^126-25 = ^3-7875. , 6733 r 3-7^ 1 25-5- X> 27750 Hence, the time required is 2 ^'5 years 3-7875 _ IC = 6-733 years = 6 yrs. 9 mo. nearly 2 55'5 = 6 years 267 days. 10-9 i-i 267 EXAMPLE. One man lent another a certain sum of money, receiving interest for the loan for 4 years at the rate 0/3% per annum Simple Interest. In all, he receives ^84. 125. 6d. as interest. What was the sum of money lent ? The interest on ;ioo for 4 years at 3 % is ^12. If 12 is the interest on ^100, ^84-625 is the interest on Ioox84 ' 62 5 SIMPLE INTEREST 285 EXAMPLE. A person, A, borrows from B a sum of money > agreeing to pay B back at the end of 2\ years a sum equivalent to both the principal and the interest accumulated during that time at the rate of 5 % per annum Simple Interest. He pays back the sum of .405. What money did he borrow ? In 2\ years, at 5 %, the interest on 100 would be ;i2^. For ^112-5 paid back, 100 was borrowed. For 405 paid back, x 4 ^ was borrowed A therefore borrowed 360 from B. EXERCISES LXIX. d. State answers in all cases true to the nearest \d. N.B. In the Exs. 1-52, simple interest is charged ; they must, however, not be regarded as illustrating practical transactions. 1. A man borrows ^875 from a bank, paying back interest every year at the rate of 5 % per annum simple interest. How much will he have paid in interest at the end of three years ? Find the Simple Interest on the following : 2. ,288. los. od. for 4 yrs. at 3 per cent, per annum. 3. ,3569. 7s. 6d. for 3 yrs. at 3^ per cent, per annum. 4. ,192. 15-5-. od. for 4 yrs. at 2\ per cent, per annum. 5. ,693. i2s. 6d. for 4 yrs. at ,3. 2s. 6d. per cent, per annum. 6. ^1756. 8j. gd. for 3^ yrs. at 2^ per cent, per annum. 7. ,167. 15-y. od. for 3 yrs. 4 mo. at 3 per cent, per annum. 8. ^5 OI - 1 9 S - 6d. for 5 yrs. at 3. 6s. 8d, per cent, per annum. 9. ^159. i6s. 8d. for 5 mo. at 5 per cent, per annum. 10- ^3584- i$s- >d. for 7 mo. at 2| per cent, per annum. H ; 6 53- J 8J. <^d. for 8 mo. at 4^ per cent, per annum. 12. .1920 for 10 mo. at 2. 15^. od. per cent, per annum. 286 A MODERN ARITHMETIC 13. A man borrows .560, agreeing to pay interest at the rate of 5 % per annum, simple interest, the latter being charged on the number of weeks (a portion of a week being counted as a whole week, and reckoning 52 weeks to the year). If he pays back the money with interest at the end of 143 days, what does he pay? 14. Under similar conditions to the preceding, find the interest on (a) ^512. 1 7s. 6d. for 33 days at 3 % per annum, simple interest. () ,792 for 45 days at 5 % per annum, simple interest. N.B. In the following examples the interest is calculated as being proportional strictly to the time, although often the time is over a year. Find the Simple Interest on 15. ,852. lew. for 146 days at 4 per cent, per annum. 16. ^1349- 7-y- 6d. for 4 yrs. 73 days at 3 per cent, per annum. 17. ,9040 for 2 yrs. 292 days at 4 per cent, per annum. 18. ,125 for i yr. 219 days at 3^ per cent, per annum. 19. In what time will the simple interest on ,732. 5^. at 2^ per cent, per annum become ,36. I2s. $d. ? 20. In what time will the simple interest on ,193. 12s. 6d. amount to ^19. js. -$d. at ^3. 6s. &d. per cent, per annum ? 21. How long will the simple interest on .83. 15^. be in amounting to 12. us. $d. at 3 per cent, per annum ? 22. The simple interest on ,491. i6s. 8d. at 2\ per cent, per annum accumulated to 61. qs. jd. ; how many years was the sum bearing interest ? 23. Reckoning simple interest at ^3. 6s. Sd. per cent, per annum, in what time will the interest on ,2742 become ^457 ? 24. In what time will the simple interest on ^93. 8s. gd. become 4. 13^. 5^., reckoning the rate to be at 2 per cent, per annum ? 25. In what time will the simple interest on ^452. 12s. 6d. amount to ,56. i is. 6%d. at ^3 J per cent, per annum ? SIMPLE INTEREST 287 26. Find the Simple Interest on : (a] ,95. i6s. 8d. for 300 days at 3 per cent, per annum. W l &5' 1 7 S - &d. f r 2 5 days at 4 per cent, per annum. (V) ,759. IQJ-. for 3 yrs. 90 days at 2. los. per cent, per annum. (d) ,653. 7s. 6d. for 5 yrs. 200 days at 3 per cent, per annum. 27. Simple interest being charged at 3 per cent, per annum, in what time will the interest on ,549. 7s. 6d. amount to ,65. iSs. 6d.l 28. In what time will the simple interest on .5319. 15^. amount to .1994. i&s. \\d. at 2\ per cent, per annum? 29. In what time will the simple interest on a sum of money be equal to the sum, reckoning interest at 3$ per cent, per annum ? 30. How much longer must ,262. los. be invested, when the rate per cent, is 3^ per annum, than when it is 3^, the interest being ,26. $s. in each case ? 31. At what rate per cent, per annum will the simple interest on ,857. i2s. 6d. become .107. 4$. of</. in 5 yrs. ? 32. At what rate per cent, per annum will the simple interest on .1927. 14,$-. become ,160. I2s. lod. in 4^ yrs. ? 33. The simple interest on ,176. gs. jd. in 3 yrs. amounted to 17. I2s. i\\d. ; what was the rate per cent? 34. The simple interest on ,200 in 4 yrs. amounted to 20 guineas. What was the rate per cent, per annum ? 35. At what rate per cent, will the simple interest on ,796. i8s. 4d. become ,99. 12.?. ^\d. in 4 yrs. 2 months? 36. At what rate per cent, will ,1957. iSs. 4^. bear ,19. us. jd. as interest in 146 days ? 37. The sum ,851. 15^. od. remained at interest for 15 yrs., when the interest had become ,319. 8s. \\d. What rate was allowed? 38. The simple interest received on ^9583. 15.$-. in 16 yrs. amounted to ,3450. $s. What was the rate per cent, per annum ? 39. A banker charged a manufacturer ^871. 17^. 6d. for the loan of -3 8 75 for 5 yrs. What was the rate per cent. ? 288 A MODERN ARITHMETIC 40. A money-lender charged a borrower is. for the loan of ,1 for i month. What was the rate per cent, per annum ? 41. The interest allowed on a deposit of ,234. 8^. ()d. for 3 yrs. 9 mo. was ,23. 8j. io\d. What was the rate per cent. ? 42. At what rate per cent, per annum will the interest on ^587. loj. in ii months become ,21. los. iod.? 43. What sum of money will yield ,23. 17^. 6d. as interest in 6 yrs. at i\ per cent, per annum ? 44. From what sum of money will ,49. $s. 6d. accrue as simple interest in 4 years at ,3. 2J. 6d. per cent, per annum ? 45. What sum of money remaining at simple interest for 5 years at 3^ per cent, per annum will amount to .512. 13^. $d. ? 46. A person borrowed a sum of money for 6 years paying 3! per cent, per annum as simple interest, and repaid the sum and interest thereon by the payment of .704. 7-y. 6d. Find the sum borrowed. 47. The simple interest on a mortgage for 8 years at ,4. 7.?. 6d. per cent, per annum was ^945. What was the amount of the mortgage ? 48. A sum of money lent for 6 years at ,3. los. od. per cent, increased to .290. 8s. od. What was the loan ? 49. A borrowed from B the sum of ^960 for 4 years, simple interest being charged at 3 per cent, per annum. It was to be repaid by one-fourth of the principal being paid at end of each of the first 3 years, and the last fourth with the interest to be paid at the end. What would the last payment be ? 50. The simple interest every half year on a sum bearing interest at ,3. 15^. od. per cent, per annum was .19. 15^. od. What was the sum invested ? 51. A sum of ;75 is annually deposited in a bank for 4 years ; if it be supposed that simple interest is obtained on the amount, what amount will be standing to the credit of the depositor at the end of the fourth year interest being allowed at 2^ per cent, per annum ? REVISION PAPERS 289 52. From what principal will an interest of ^52. los. od. accrue in 7 years, simple interest at 5 per cent, per annum being obtained ? 53. In the following extracts from various advertisements, find at what rate per cent, per annum, simple interest, the purchaser is charged if he chooses the instalment system in preference to the cash : (a) Sewing machine, price 49^. 6d., or 1 1 monthly payments of 5.9 (fr) 9-ct. gold chain, 30^. ; $s. monthly, or 2js. cash. (c] Cash, ^5. 5^. ; or credit, los. 6d. down, and 10 monthly payments of los. 6d. (d) Cash price, 55^. ; or deposit 6s., and nine monthly payments of 6s. (<?) Piano ,32. cash, or 36 monthly payments of 2 is. (/) Billiard table, 5 ft. 4 in. by 2 ft. 10 in. Cash price, ^5. 15^., or 13 monthly payments of los. 54. At what rate per cent, per annum is simple interest being charged for credit in the following cases : (a) " For students commencing their medical studies, entrance fee, 30 guineas. The entrance fee is payable by all students at the commencement of their studies, or may be paid in three instalments of twelve guineas at the beginning of the first, second, and third years." (&) Pawnbroker's advertisement. "The use of 10 for one month costs 2s. 6d." REVISION PAPERS. [Compiled mainly from Civil Service and other Examination Questions.] A 1. Each of nine loaded trucks weighs 7 tons 12 cwt. What is their combined weight ? 2. A sum of ;n. us. is to be divided equally between 8 persons, how much will each receive? 3. . How many centilitres are there in 37 litres 4 centilitres ? 4. How many yards are there in 400 inches, and how many feet and inches over ? J.M.A. T 290 A MODERN ARITHMETIC 5. A jug which holds pint is filled from another containing 2| pints. How many pints are left in the latter ? 6. Each of three measures will hold 2^ Ib. of butter. How many Ib. of butter does it take to fill two of these and half fill the third ? 7. I have four weights of 3-5, 2-75, 1-25,0-05 kilograms. How many kilograms do they weigh together ? 8. Express in pounds and decimals of a pound what is left after paying away ,4-376 out of 6. 9. Find the area (in square feet) of a plank 6-35 ft. long and 0-66 ft. wide. 10. How many times can a bottle which holds 0-125 gallon be filled from a cask containing 8-5 gallons? 11. What fraction of a ton is 300 Ibs. Give your answer in its lowest terms ? 12. Express as a decimal the ratio of 21 pence to a . 13. Express f of a gallon in pints to the nearest pint. 14. What percentage of an hour is 3 minutes. 15. Find the value of 3 per cent, of 8. Neglect fractions of a penny. 16. Find in feet and inches the total length of 50 planks, each 1 1 ft. 9 in. long. B 1. What is the least fraction which, added to the sum of f | and f f , will make the result an integer ? 2. It requires 1344 tiles, each 9 in. by 4^ in., to cover a court-yard. What is its area ? 3. A and B own a field in shares proportioned as 15 to 13. If A's share is f of an acre, what is the size of the field in square yards ? 4. What decimal of n cwt is 3 qrs. 21 Ibs.? ot what decimal of 10 kilograms is 75 grams ? REVISION PAPERS 291 5. Find the product of 0-0119 an d 2-967. Divide this product by 21-93. 6. What amount of interest does ^1020 yield per annum, if invested at 2| per cent. ? 7. Find the least common multiple 01 78, 84 and 90. 8. Three cyclists who are riding together have machines with wheels, respectively, 78, 84 and 90 inches in circumference ; what is the least distance in yards that they must travel in order that their wheels shall be simultaneously in the same position as at starting ? 9. Find the decimal equivalent to the fractional expression 10. A box 4 feet long, 2 feet wide, and i feet deep (internal measurements) weighs 10 Ibs. If it be filled with water weighing looo oz. for each cubic foot, what is the total weight of box and water ? C 1. A square field is 15 ac. 2 ro. 15 per. in area ; find to the nearest foot the length of a path crossing it diagonally. 2. Given that a French metre is 39-37079 inches. Show that the difference between 5 miles and 8 kilometres is very nearly 51 yds. 3. Express in tons 203,549 ounces and 93 m. 660 yds. 2 ft. in inches. 4. A hogshead full of sugar weighs 13 cwt. I qr. 5 lb.; the hogshead itself weighs 2 qrs. 19 lb. How many packets of sugar each weighing 2^ lb. can be made from the quantity, and what amount of sugar will be left ? 5. I buy 500 oranges at two for three halfpence, and another 500 at two for three pence, after which 60 of the better sort are eaten. I then sell the remainder at five farthings each. How much do I gain or lose? 6. The cost of 5 cwt. 2 qrs. 14 lb. of coffee is ^56. i$s. yd. If it be sold at is. i \\d. per lb., what is the gain or loss ? 29 2 A MODERN ARITHMETIC 7. A merchant imports 2 tons of tea at 4. los. per cwt. He pays ^10. 6s. Sd. for freight and dock charges and a duty of 4d. per Ib. He sells the tea at 7. 5* gd. per cwt. Find his gain per cent. 8. A woman buys eggs at 5 for 3^. and sells them at 3 for 2d. How many eggs must she sell to gain is.? 9. By selling an article for ^9. 12s. 6d., a tradesman loses 3f per cent. At what price should he sell it to gain 2^ per cent.? D 1. A clock loses 8-5 sees, an hour when the fire is alight, and gains 5-1 sees, when the fire is out, but on the whole it neither loses nor gains. How long in the 24 hours is the fire burning ? 2. If a farmer pays ,33. 15^. as rent for 27 acres, how much should he pay as rent for 70 acres ? 3. Find the value in minutes and seconds of 8 per cent, of 4 hrs. 20 min. 25 sees. 4. A farmer has a flock of 820 sheep ; of these 123 die and the rest are sold. What percentage is sold ? 5. How many sq. ft. of paper will be required for the four walls of a room 16 ft. 8 in. long, 12 ft. 6 in. wide and 12 ft. high? 6. If the wages of 12 men for 210 days be ^514. ios., how much ought they to receive for 180 days' work? 7. Find the value of 15 per cent, of ,96. 17.9. 6d. 8. How much per cent, of 2 cwt. is 7 Ib. ? 9. What fraction added to the continued product of f , 2}, 9%, 2 T 6 T will give the continued product of |, 9, , i|| and ? 10. A dealer purchases goods and sells them at a profit of 1 2^ per cent, on the cost price ; how much does he receive for each of his outlay ? 11. Income tax being 8d. in the , how much per cent, does this represent of a man's income ? REVISION PAPERS 293 E 1. Simplify * * 3* of a* -6* of 3* of * 7f of 3* 2. Divide 3-425 by 0-002192. 3. What principal, if invested for 3 years at 2f per cent, per annum simple interest, will amount to ^575. ioj. yd. ? 4. Find the value of 2} of 3,} of 4 Ibs. 8 oz. 10 dvvt. 12 grs. Troy. 5. Reduce ,4. iSs. io\d. to the decimal of 5 guineas. 6. Find the cost of 276 tons 16 cwt. at ,3. iSs. n^d. per ton. 7. Find the least common multiple of 385, 231, 165, 105. 8. A room 21 ft. 4 in. long, 18 ft. 8 in. wide, and 15 ft. 6 in. high, is papered with paper 32 inches wide at one shilling a yard. What is the total cost ? 9. If by selling a certain horse for ,66 I should lose 28 per cent, of the cost at which I bought the animal, what is my loss? 10. Prove that the product of any two numbers which consist of three figures and four figures respectively must be a number consisting of not less than six or more than seven figures. F 1. Find the cost of 49 cwt. 3 qrs. 14 Ibs. at ^13. &s. lod. per cwt. 2. Divide 5-4 by 0-00072 and the quotient by 1,470,568 to four places of decimals. 3. Find the greatest common measure of 27531 and 8740. 4. Add &+& + &+& + & 5. Find (in its lowest terms) the difference between ff and JUJL 191T- 6. The French metre is 3-281 English feet, the Rhenish foot is 12-356 English inches. Express 4-567 metres in terms of Rhenish feet to 3 places of decimals. . 294 A MODERN ARITHMETIC 7. A sum of money put out at simple interest amounts to ;688, when the rate is i\ per cent, and the time 3 years. What would the amount be if the rate were 3^ per cent, and the time T.\ years ? 8. Show that if a square number ends in 6 the figure in the tens place is odd ; if it ends in any other number, the tens figure is even. 9. If B's wages are ^ of A's, for how many days will a sum which pays A's wages for 119 days pay the wages of both together ? 10. If the manufacturer makes a profit of 25 per cent., the agent one of 8 per cent, and the shopkeeper one of 20 per cent., what is the cost to the manufacturer of an article which is sold in the shop for ^32. 8s. ? 11. A man buys an article and sells it at a profit of 10 per cent. If he had bought it at 10 per cent, less, and sold it for 6d. less, he would have made a profit of 20 per cent. Find the cost price. 12. To do a piece of work, a contractor can employ two classes of workmen, whose wages are in the ratio of 17 to 13. If he employs the higher paid men (who work the faster) he pays 148. i$s. in wages, being 10. los. less than the lower paid men would cost him. Compare their rates of work. 13. A milkman adulterates his milk as follows : He takes a pint out of each gallon and replaces it with water. He then takes another pint out of the mixture and replaces that with water ; and he repeats the operation a third time. If he then sells his adulterated milk at the price per gallon that the pure milk origin- ally cost him, what is his gain per cent, on the selling price? 14. A tradesman makes up his receipts monthly. His receipts for the first two months average ^3 less than those of the first month ; those of the first three months average ^4 more than the average of the first two ; of the first four $ more than that of the first three ; of the first five 2 less than that of the first four ; and of the first six 7 more than that of the first five. If the REVISION PAPERS 295 average monthly receipts for the first six months are ^611, what were the receipts for each month ? G 1. Find a decimal of 3 figures as nearly equal to as a decimal of 3 figures can be. 2. Find, to the nearest litre, the content of a tank 2-34 metres long, 1-23 metres broad, and 45 centimetres deep. A litre is a cubic decimetre. If these dimensions are only true to the nearest centimetre, find the greatest possible error in the result. 3. The population of the central area of London was 821,257 in 1891 and 753,260 in 1901. What is the percentage decrease during this period of 10 years, and, if the rate of decrease during the next 10 year period is the same, what will be the population in 1911 ? 4. In 1901 Ontario had 1,278,635 acres under wheat, which produced 21,515,780 bushels ; Manitoba had 2,011,835 acres under wheat, which produced 50,502,085 bushels. Find out which pro- vince produced the heavier crop, and give for each (to the nearest whole number) the number of bushels to the acre. 5. A man borrows ^100, to be repaid by instalments, and agrees to pay 4 % interest on what he owes at any time. He pays off ,30 (including interest) at the end of each year, until a year comes when a less sum than ,30 pays off the debt. What is this 6um, and what is the total of the repayments ? 6. A bar of iron 10 feet long has a rectangular section shown below. Measure the length and breadth, and find the area of the section. Also find the weight of the bar, a cubic foot of iron weighing 490 pounds. 296 A MODERN ARITHMETIC H 1. A, B and C go into partnership, A investing ,4000, B .3000, and C ^2000. If the profits for the first year are ,1530, how must they be divided ? 2. Write down all the prime numbers between 50 and 100. Can you give any reasons why not more than two of these prime numbers are consecutive odd numbers ? 3. A parliamentary grant is made at the rate of 5^. per head for all the children at elementary schools. If this grant is distributed at the rate of 5^. qd. per child in town and y. $d. per child in country schools, what percentage of the total number of children are in each class of school ? 4. A batsman has a certain average of runs for 16 innings. In the 1 7th innings he makes a score of 85 runs, thereby increasing his average by 3. What is his average after the I7th innings? (There are no "not out" innings.) 5. A bicyclist rides up a hill 2 rniles long in 9 min. 20 sec. and then down a hill of the same slope 3 miles long in 10 min. 30 sec. How long will it take him to ride back, assuming the speeds to be the same in each case, whether the hill be long or short ? 6. How many postage stamps, each Jf inch long and f inch wide, would make a frieze a foot wide all round a room 1 6 ft. 6 in. long and 14 ft. 9 in. wide? 7. A cistern is full of water. A cylindrical brass piston 16" long and 10" in diameter is immersed in the cistern. How many cubic inches of water will run over, and how many gallons would the overflowed water fill ? 8. There is a square lawn ABCD containing i ac. 785 sq. yds. How much shorter is the diagonal AC than the sum of the two sides AB, BC? 9. Find the area of a rectangular lawn 55 ft. long which measures 170 ft. all round. 10. A shallow rectangular dish is 0-36 metres long, 0-25 metres broad and 0-4 cms. deep. Express in grammes the weight of water it will hold. PART II 49. MEASUREMENT OF LENGTH (continued}. EXERCISES on the use of the ordinary foot-rule and metre scale have been given already, and also on the use of calipers and other simple means of measurement; in the following exercises additional apparatus is necessary. EXERCISES LXX. Practical. Use of Screw-gauge and Spherometer. [Apparatus: Micrometer Screw-gauge. Spherometer. Scale. Glass slips. Wires of different diameters. Glass rods. Balance] 1. Examine a Micrometer Screw-gauge (Fig. 38), note the distance the screw moves forward or backward for I, 2, 3, etc., complete revolutions of the screw head. 2. Find the reading when the screw is in contact with the lower jaw. Open the jaws, FlG . 38 . -Micrometer Screw-gauge. and introduce a thin glass slip between them ; screw up the jaws and take a new reading. 3. Introduce a second slip between the jaws. Find the reading when the jaws, are both in contact with the slip : open the instrument and re-introduce the slip used in Ex. 2 on the top of the second slip : find the new reading. 298 A MODERN ARITHMETIC 4. By experiments similar to Exx. 2 and 3 note that, when the screw head is turned through equal angles, the screw is moved forward or backward through equal distances, and that the methods of proportion may therefore be adopted in calculating the distance moved forward by the screw. 5. Measure the diameter of the given wires with a screw-gauge. 6. Check the measurements as follows : Wind the wire carefully round a lead pencil or glass tube, taking care that each turn of the wire touches the previous one at all points. Make 20 turns, and measure the distance from the outside of the first turn to the outside of the twentieth turn by means of a millimetre scale. Divide this measurement by 20, and so obtain the diameter of the wire. Repeat the experiment with varying numbers of turns of wire. 7. Measure the thickness of the given rectangular glass slips. FIG. 39. Spherometer. 8. Measure the length and the breadth of any one of the slips used in Ex. 7 ; calculate its area. Next, find its volume by the balance or by displacement : divide the volume by the area, and so calculate the thickness. Compare your result with the result in Ex. 7. MEASUREMENT OF LENGTH 299 The principle of the spherometer (Fig. 39) is precisely the same as that of the micrometer screw-gauge. 9. Measure the thickness of the given plates by means of the spherometer (Fig. 39). Check your results by using the screw- gauge. EXERCISES LXXI. Grapnic. Use and Principle of Vernier. 1. On squared paper construct a scale of inches, mark it so as to show 10 inches, cut out another length similarly divided, and from the two make a single scale 20 inches long. Mark off again a length of 9 inches (10-1) on another piece of squared paper, divide this up into 10 equal parts, number these o, i, 2, ... up to 10. This will serve as a vernier. (a) Note the difference between I scale division and i vernier division, between 2 scale divisions and 2 vernier divisions, and so on. (<) Slide the vernier until the division marked i on the vernier coincides with the division marked i on the scale, note the distance between the zero marks, slide the vernier until the divisions marked 2 coincide, again note the distances between the zero marks, and so on. 2. Set the vernier so that the distance in inches between the zero marks is (a) 0-3, 0) 0-6, (c) 1.2, (d) 3-5, (') 5'6, (/) 4-2. 3. Draw a scale with its main divisions inches, and the sub- divisions tenths of an inch ; construct a vernier attachment to read to o-oi of an inch. 4. Construct a scale and vernier to read to 0-02 of a cm. 5. Draw a line 7-2 inches long ; divide it up into 8 equal parts ; alongside attach a vernier, in which a distance equal to the 300 A MODERN ARITHMETIC scale has been divided up into 9 equal parts ; what fractions of the whole line are there between corresponding divisions of the main scale and the vernier attachment, when the commencements of both scales are exactly opposite ? 6. Attach to an ordinary scale of tenths of an inch, a vernier suitable for reading to the nearest 2ooth. EXERCISES LXXII. 1. A screw has its pitch such that, on turning the screw 8 times round, its end moves through a distance of 1-84 in. How many times must it be turned round in order that its end may move through a distance of 0-8602 of an inch? j\rg Th e pitch of a screw is the distance between successive threads, or the distance the screw moves forwards or backwards during one complete revolution. 2. A spherometer (Fig. 39) consists of a screw, the end of which moves forward through a distance of 0-5 mm. for each com- plete revolution of the screw. What distance corresponds to a turning of the screw equal to 0-265 f a revolution ? 3. In lifting weights by means of a windlass, the men have to move round through a distance of 25-12 ft. in order that the weight may be raised through a distance of 5-76 ft. What distance must the men move through, in order that the weight may be raised through a distance of 370-8 yd. ? 4. A block of pulleys is used to raise a weight, and it is found that when the weight is lifted through 3-52 ft. an amount of rope 24-64 ft. in length has to be pulled in. What distance will the weight be lifted through when 61-25 ft. of rope are pulled in ? 5. The graduated rim of the moving screw of a gauge (Fig. 38) is divided into 50 equal parts, and it is found that when the head is moved forward through a known distance of 0-542 cm. the screw has been turned through 10 complete revolutions, together with an additional 42 divisions. What is the pitch of the screw ? 6. A boy made a rough spherometer as follows : He first cut a circle in cardboard, and divided the circumference of the circle MEASUREMENT OF LENGTH 301 into 32 equal parts. He then attached it to the head of a screw the pitch of which was 0-05 in., and afterwards suitably mounted it. The thickness of a pertain slab corresponded to 2 complete turns, and a further motion through 16 divisions. What was the thick- ness of the slab ? 7. If the head of a screw moves forward through 0-24 in. when turned through 4-25 revolutions, what should it be turned through in order to move forward through a distance of 0-33 of an inch ? 8. The pitch of a screw is known to be 0-08 in. What is the pitch of another screw which requires 6-24 revolutions to- move forward through the same distance that the former screw moves through in 10-92 revolutions, and what is the distance ? 9. The pitch of the screw of a spherometer is 0-5 mm., the diameter of the rim of the graduated circle is 4 cm. What distance will a point on the graduated rim be moved through in measuring a thickness of 2-36 cm. ? 10. The diameter of the graduated rim of a spherometer is 2 in., and it is known that in measuring a distance of 018 in. any point on the rim has been turned through a distance of 22-608 in. What is the pitch of the screw of the spherometer ? 11. A hockey stick has a cylindrical handle, which is wound as thickly as possible with thread. What is the diameter of the handle, and what is the diameter of the thread, if 450 turns of thread are required to cover the handle to a distance of 18 in., and if the length of thread used is 177 ft. ? 12. When used in estimating a thickness of 1-6 mm., the reading of a spherometer is 8 on the scale, and 20 divisions on the graduated circular rim : when used in estimating a thickness of 2-46 mm., the reading is 12 on the scale and 45 on the rim. What is the thickness, if the reading is 13 on the scale and 30 on the rim ? (The total number of divisions on the rim is 100 and the distance between the marks on the scale is equal to the pitch of the screw.) 13. Two similar millimetre scales are engraved on two flat rods A and B of different metal, both being at the temperature at which 3 02 A MODERN ARITHMETIC ice melts ; the rods are heated to the temperature at which water boils, and it is now seen that the scale divisions of one rod A are bigger than the scale divisions of B, and that 132 divisions of A seem to coincide with 133 of B. The divisions of B are 1-0018 times as long at the temperature of boiling water as they are at the freezing point. What is the new length of a portion of A which was i metre in length at the freezing point ? 14. A line AB is divided up into 36 equal parts by 35 marks, numbered in succession i, 2, 3, ... 35, while A is marked o and B marked 36. Another line ab has a length of 35 divisions, it also is divided into 36 equal parts and numbered similarly to AB. If the two ' scales ' are placed along- side one another so that A and a coincide, what is the distance between the 29th marks on the two scales, when expressed as a frac- tion of the length of AB? If ab be moved along AB so that the 24th division of ab coincides with the 24th division on AB, what is the distance between A and a ? 15. The right-hand side of Fig. 40 shows a portion of a barometer scale with vernier attachment, in which it will be noticed that each inch on the scale is divided up into 20 equal parts, and that 25 divi- sions of the vernier are equal to 24 of the scale. To what degree of accuracy can readings be taken ? Explain the vernier attachment FIG. 40. shown on the left-hand side of Fig. 40. 16. Assuming that in dividing up a scale, it is not expedient to have divisions closer together than the twentieth part of an MEASUREMENT OF LENGTH 303 inch, show how to divide up a scale of inches and a suitable vernier in order to read true to (a) 2 foy in. () 2 <y in, (c) T h" in - W ?fa in - (e) ^ in. (/) ihf in. 17. A piece of metal (a) 09 gram, (b) i-i gram, is broken up into 10 equal pieces ; explain how you would use them in conjunc- tion with decigrams, to weigh to the nearest centigram. 18. The swinging pendulums of two clocks are viewed through a telescope, so that one appears to pass directly behind the other. The pendulums both appear vertical at 12 o'clock ; they next appear vertical together at 8 minutes past 12 o'clock. If one clock beats seconds, find the time which elapses between the 25oth passages since 12 o'clock ; what also is the time of swing of the pendulum of the second clock ? 19. A person observes that the pendulums of two clocks cross the vertical together every 5 minutes. If one clock be correct, what does the other gain or lose in one day ? 50. THE USE OF THE CHAIN. The ordinary chain used in surveying, and known as Gunter's chain, is 66 feet long. It is divided into 100 links, joined together by small rings. To every tenth link is attached (Fig. 41) a piece of brass (sometimes called a finger) in order to facilitate the counting of the links ; and FlG - brass handles are attached to the ends of the chain for pulling it tight. Many small surveys can be made with a chain only; larger surveys require the assistance of the theodolite and other instruments. 34 A MODERN ARITHMETIC EXAMPLE. To survey the small field shown in Fig. 42. (a) Four main survey lines are obtained as near to the boundary of the field as possible ; suppose these are AB, BC, CD, DA; the lengths of these lines are found by the chain. FIG. 42. The lengths of the two diagonals AC, BD are also found (one diagonal is really sufficient, but it is generally a safer plan to measure both, one serving as a check for the whole survey). Offsets are then taken along these and probably other lines, thus : Suppose P, Q, R, S, T (Fig. 43) are more or less important or salient points on the boundary in the neighbourhood of AB (line i). Measurements are taken along AB from some convenient starting point O until a point p is reached, such that /P is at right angles to OAB, the distances O/ and />P are noted, say 500 links and 250 links ; similarly measure- ments of Cty, ^Q ; Or, rR ; etc., are taken, and the readings set down in the ' field ' book. The entry might be somewhat as follows : Line i 80 1500 100 I2OO 1000 350 780 250 500 120 O A p q t B FIG. 43. THE USE OF THE CHAIN 305 The readings mean that O/= 500, /P= 250, and P is to the left of OAB. 0^=780, ?Q = 350, and Q Or= 1000, rR= 120, and R is to the right of OAB. O5- = 1200, ^8= ioo, and S ,, left 0^ = 1500, /T= 80, and T The figures express the distances in links. Similarly the form of the boundary in the neighbourhood of BC, CD or DA is found. EXERCISES LXXIII. Graphic. 1. Fig. 44 represents a railway, fence and stream, scale i chain to the inch. Make a field-book entry with reference to the line AB. FIG. 44. 2. Trace the form of a portion of a fence from the following field-book entry : IOOO o fence 800 25 700 20 fence o 500 10 250 15 1 80 fence o 120 J.M.A. 306 A MODERN ARITHMETIC 3. Draw to scale the perimeter of a pond from the following readings, taken with reference to three lines forming a triangle within which the pond is situated. End of Line i. 2000 -^ Line 3 1500 150 1 200 IOO IOOO 30 700 30 300 250 O <* Line 2 End of Line 2. End of Line 3. Line 3*- 2000 Line I sr 2000 40 1900 3o 1700 1 60 1500 So 1500 170 IOOO 60 IOOO 180 400 50 500 200 200 20 300 Line I ^ Line 2 X 4. Draw a plan of a field from the following field readings : End of Line i. 1810 | -^ Line 2 160 1800 160 IOOO 140 350 130 300 120 ^ Line 3 End of Line 2. End of Line 3. hedge 1800 ^ Line 3 200 2000 >s. Line I 230 1750 280 1500 220 1200 30 1300 hedge) 800 300 IOOO 240 590 150 600 160 3 80 50 500 300 200 30 200 ! ^ Line i O ^ Line 2 THE USE OF THE CHAIN 307 5. The following plans are drawn to scale ; make field-book entries along the lines i, 2, 3, 4, 5 for the points indicated. (a) Fig. 45. (6) Fig. 46. FIG. 45. (Scale i chain to the centimetre.) FIG. 46. 51. APPROXIMATIONS (continued). Approximations in Multiplication. In many cases calculations have to be performed in which the terms involved are very nearly convenient round numbers, but not quite. Consider the following examples : EXAMPLE i. Evaluate 127x96. 127x96=125x96 (approx.) = 1 2000. Accurately 127 x 96 = 96 x (125 + 2) = (125 x 9 6) + T f5(i25x 96)= 125x96 + 3^x96x125 = 125 x 96 increased by 16 in every 1000 or by 1-6 % -12000 = 12192. 3 o8 A MODERN ARITHMETIC EXAMPLE 2.-- Evaluate 253 x 484. 253 x 484 = 484 x (250 + 3) = 250x484 increased by 3 in every 250, i.e. 1-2 % or 12 in each 1000 _ ipoo v J.Q. = x x 44 ?> ?) '5 = 121000 ,, = 122452. EXAMPLE 3. Evaluate 48-1 x 12-6. (48-i)(i2-6) = (48 + o-i) x (12-5 = 600 ( i + ^ + T i T + T i T x T J^), neglect j^ x T J y . .'. ( 4 8-i)(i2-6) = 600 + 600(^0) + 600 ( T i ? ), that is to say, since x^ is roughly 0-2 %, and y-lrg is 0-8 %, we increase 600 by (0-2 4- 0-8) %,/.*. by i %, and an approximate value is 606. Actually multiplying 48-1 by 12-6, 606-06 is obtained. 48-1 12-6 481 96-2 28-86 606-06 A rough approximation should be mental or nearly so, and is chiefly of use when the numbers to be multiplied are large. EXAMPLE 4. Evaluate 497-5 x 600-6. The terms are 2-5 in 500 less than 500, and 0-6 greater than 600, that is 0-5 % less than 500 and o-i % greater than 600. A first approximation = 500 x 600 = 300000. ' APPROXIMATIONS 309 A second approximation increases this by (o-i -0-5)%, i.e. decreases by 0-4%, i.e. 4 in every 1000. .'. second approximation = 298800 (accurately 298798-50). EXAMPLE 5. Evaluate 7493 x 798-2. (7493)(798-2)= 7500 x 800 nearly = 6000000 approx. This has to be reduced by 7 in 7500 and also by 2 in 800, i.e. by 0-26 % approx. Hence 7493x798-2 = 6000000 reduced by 0-26 % approx. = 5984400. Multiplying accurately 5980912-6 is obtained. 7493 798-2 5245 1 67437 59944 1498-6 5980912-6 The method is more particularly useful when several terms have to be multiplied together. EXAMPLE 6. Find the weight of a block 127-6 cm. long, 79-3 cm. broad, 60-9 cm. thick and having a density of 11-22 gr. per c.c. The weight = (127-6) x (79-3) x (60-9) x (11-22) grams. A first approximation is 125x80x60x11 = 6600000 grams or 6600 kilo. A nearer approximation increases this by (2 - 0-9 + 1-5 + 2)%, i.e. 4-6%, or about 3 in 66, and a second approximation is 6900 kilo. Approximations in Division. It has been shown already (p. 45) that the area of a rectangle with sides i + x and i - x is equal to i - x 1 , that is, (!+*)(! -*) = (!-**); A MODERN ARITHMETIC i - x , and i+x I - i + x i x Accordingly, if x be small, so that x 2 - can be neglected, i+x \-x and i -x + X. Or, dividing by ( i + x) is practically equivalent to multiplying by i-x, and dividing by (i-x) is practically equivalent to multiplying EXAMPLE i. Evaluate approximately ^ 9' 99 7%\32 7 89-32 99 In other words, divide by 100 instead of 99 and increase by i %. EXAMPLE 2.-Evaluate 792 IO2 X 297 x 792 io 4 x 800 as a first approximation. 102 x 297 ioo x 300 But since 9835 is approx. 2 % less than 10000, 792 i % . 800, 102 2 % greater ioo, 2 97 i % less 300. 9835 x 79 2 _ Io4 x 800 102 x 297 ioo x 300 _ ^ \ /O /O /O /O/ approx. = 266 increased by ( - 4 %) = 256 as a second approximation (actually = 257). APPROXIMATIONS IN DIVISION 311 When generalized, the method used in the above example may be written as follows : if x, y, z, w are small fractions, either positive or negative. So also N(i+^) n =N(i +nx), EXAMPLE 3. Evaluate (8%) = io 2 x -^ - approx. = i53 x io 2 . 8^2 3 By straightforward evaluation, "2 = 15-4 x io 2 . EXERCISES LXXIV. a. Mental or Oral. Decimalize approximately : 1. . 101 5. 736. 99 q 1 120 9 ' "808- 13. 756. 248 9 962 3. 5636 202' 4. 5978 206' ^J - 102 fi 8l6 6> W 7. 3624 198' 8. 2?. 102 10 224 792' 11, 144 + 3 72 + 3' 12. 596. 152 14 860 15. 98 x 103 (I02)(l03) 101 X97 312 A MODERN ARITHMETIC Decimalize approximately : 16. 868x216 430 x 440' 18. (I-02) 3 ' 19 97 * 42 -Lt7. - X - "7* 32 256 20. 29.7 EXERCISES LXXIV. b. Fill up approximately the missing parts of the following tables. (Assume that the circumference of a circle = 3- 141 59 x diameter.) 1. Diameter of circle. Circumference of circle. (a) 9-8 ? <*) p 63-46 fc) 25-1 p (d} 33-3 ? W 50-5 p Mass in grams. Vol. in c.c. Density. (a) 2 5 8 6 4 ? <*) 1936 241 ? W p 103 5-02 (*0 1821 p 8-96 w 344-6 p 8-53 Amount of money. Rate per cent. Time in years. Present value. 3. (a) ^82. 6^. 4 2 p (*) ^60 2 3 p W ^120 4 I ? (rf) ^36 5 I p APPROXIMATIONS IN DIVISION 313 In the evaluation of the following exercises, it is found that the figures marked with an asterisk should have been changed as indicated. What difference approximately will this make in the 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. EXAMPLES. Error in Statement. Error in Result. Worked Examples (a) 989x47 = 46483 (d) 2791-490-=- 3-98 = 708-5 9 instead of 8 8 4 o-i %, or 47 too large. i %, or 7 too small approx. 71656-228 -=-95-06 = 753-8 5 instead of 4 p 71656-228-^95-06 = 753-8 5 6 ? 882-54019-=- 9070-3 =0-0973 7 8 p 882- 540 1 9 -=-9070-3 = 0-00973 3 6 p 594-426842 -=- O-O734 = 809-63 4 7 ? 150-3x89-7=13481-91 3 9 p 21-84-^-4-16=5-25 6 2 p 23-6 x 18-5 x 14-8 = 6461-68 8 9 p 431-25^(12-5 x7-5) = 4-6 5 6 p 7056 -=-112 = 63 56 42 p 3-142 x 40-4= 126-9 approx. 4 6 p 1 89-4 -=-3- 1 42 = 60-3 4 7 p (3-i4i)(6i-2) 2 -=-4 = 294i 2 5 p (3-i4i59)(8o-i) 2 -:-4 = 539,, i ,, 9 p v/656i = 8i i 7 p 314 A MODERN ARITHMETIC Approximations in the Extraction of Square Boot. EXAMPLE. Find the square root of 98176352-876123..., where the number is only approximately known, that is, where the decimal figures beyond the i, 2, 3 are unknown. Write a, b, c, ... for the unknown digits, and proceed to extract the square root in the ordinary manner. 9908-3980983873 ^ i ", i i ^ i I i i ~, i I 71 I i . _ i T 9 189 19808 98 i7|63|52|-8 7 |6i|23|^|^ ef\gh\kl 81 1717 1701 \op 166352 158464 19816-3 19816-69 19816-788 7888-87 5944-89 1943-9861 1783-5021 1 160-48 1 I58-53 160-484023 -534304 19816-788 1-949719^3 Still follow the ordinary method, but indicate the figures in the remainder which are incorrect, and accent the figures on the left, when, on account of the unknown digits a, b, c, d, their effect is invalidated. No. of effective figures = 6. 19816-79609 = 5. 19816-796188 = 4. 19816-7961963 =3- 19816-79619668 19816-796196767 19816-7961967743 Up to the present all the figures have been supposed accurately known, but now a, b, ... represent figures known to exist, but the values of which are unknown. Next digit is o. Period c, d is brought down. Next digit is 9, but effect of last four digits is invalid- - ated owing to a, b, c, d e f being unknown. Digit is 8, but although one di g il is added on left ' efiect f 7 wil1 be invalidated owing #m to*,/. 1 '9497 i 7835116481 i662o8x 1 58534369504 594503885889 y I729xxx 15853436957244 1 44 xxxxxxxxx xfiff 138717573377369 APPROXIMATIONS IN SQUARE ROOT 315 The process cannot be carried any further as all the figures are now unknown, the answer being 9908-3980983873. The last seven digits have been obtained without a know- ledge of the figures to be brought down. Since ineffective figures are useless, the abbreviated process may be set down as below : 9 9 o 8-3 980 98387 3 i8g| 17 17 19808 | 16 19816-3 19816-69 19816-78 63 | 52 | -87 61 \2$ \ a.b \ c .d\ ... | ... [ ... | ... | ... 6352 -000 O_ 7555-87 1943-9861 160-484023 The digit o comes next ; this she 19816-7 19816- 1981 198 19 I 1-949719 1-949719 -166208 7674 1729 ,44 added to the left, but two figures on left are invalidated owing to a, b\ the 8 on the left is therefore struck out. The second 8 on the left is now struck out, the digit in the root being 9. It will be noticed that all this abbreviated process is really that of contracted division ; and that, having obtained any number of figures in the ordinary way, an equal number of figures can then be obtained by employing the methods of contracted division. Also, in finding the square root of any number, say >/N, having had part of the answer, say , the N - n 1 remainder square root is approximately n-\ , that is, n-\ Surds. Already it may have been noticed that many vulgar fractions can only be represented as decimals by using an infinitely large number of figures, thus ^ = 0-333... and is represented by 0-3. 316 A MODERN ARITHMETIC There are other quantities which cannot be represented exactly either by a decimal or a fraction. For example, consider \/2. Y the square of i is i and 2 is 4, clearly \/2 is something lying between i and 2, that is, \>abc... , where a, fr, c, ... are certain digits. These figures cannot ter- minate. But suppose they did terminate, and that c was the last digit. Go through the process of squaring i -abc. i-abc i-abe \-abc (suppose z the last digit) 2-000 oo The figure z must be a zero, and therefore CKC must give a number ending in o, but the only possible values of c* are i, 4, 9, 16, 25, 36, 49, 64, 81. Hence the decimal does not terminate. Again, \/2 cannot be a vulgar fraction, say , for then q^2 would =/, where/ is an integer, and 2 . q* would =/ 2 . In finding the square root of an exact square (/ 2 ) we can find the factors in pairs ; but on the left we have a certain number of pairs of factors, and an odd one ; .'. \/2 cannot be a vulgar fraction. A quantity of this kind, that is \/2, is called a Surd. There is no difficulty, however, in evaluating surds to any required degree of accuracy. EXERCISES 317 EXERCISES LXXIV. c. 1. A body falling freely from rest describes a distance in feet equal to the product of the square of the time in seconds and a factor 16-1. Find the time in sec. taken by a body falling through a vertical distance of (a) 28140 ft., (b) 580 ft., (c) 220 yd., (d) 400 yd. 2. In Ex. 1 what would be the factor corresponding to 16-1 in the metric system, where the distance is expressed in centimetres ? 3. If the velocity in feet per second acquired by a falling body is numerically equal to the square root of the product of 64-44 into the height (in ft.) fallen through, find the velocity acquired in a fall of (a) 120 ft. exactly, (b) 324 ft., (c) | mile. 4. Two ships start from a lightship. One sails due North for 5 hours, at an average rate of 8*4 miles per hour ; the other sails due East for 5 hours at the rate of 17-5 miles per hour. How far apart are they at the end of the 5th hour ? 5. The length of an incline is 441 yards, and it has an average rise of 4 in. per yard. What is the height of the upper end over the lower? 6. The bottom of a mine is reached by two inclines, the first 234 ft. long, with a fall of 3 yd. for every 25 yd. along the incline; the second 516 ft. long, with a fall of I yd. in every 6 yd. along the incline. What would be the shortest distance between the top of the mine and the point where the lower incline meets the bottom ? 7. Two houses are directly opposite one another, the road between them being 37-6 ft. wide ; the bottom of a window in one house is 8 ft. higher than the bottom of a window opposite it in the other house. What is the distance between the two ? 8. The area of a room, the length of which is half as much again as its width, is 216 sq. ft. What is its width ? 9. A garden plot, the length of which is 8 times its breadth, has an area of 4418 sq. yd. What are its dimensions ? 3 i8 A MODERN ARITHMETIC 10. Find the side of the base of a square prism, the height of which is 11-2 in. and which contains 596-848 cub. in. of matter. 11. A cistern with a square base is 2 ft. 9 in. high, and has a capacity of 14 cub. ft. 1680 cub. in. What is the side of the base ? 12. A water tank with a square base is 4 ft. 7 in. high ; it can hold 1000 gals. Assuming a gallon to be equal to 277^ cub. in., what is the side of the base, true to the nearest inch ? 13. In the extraction of a square root the result obtained without any contraction is 862-5..., and the remainder is 62-03. State the next two figures in the square root. 14. (a) What is the length of the diagonal of a rectangle with sides 15 in. and 8 in.? (b) What is the length of the diagonal of a square, the side of which is 43 ft.? (true to the nearest foot). {c) How much shorter is the diagonal of a rectangle than the sum of its two sides, when the sides are 35 ft. and 69 ft.? (d) The foot of a ladder 28 ft. long stands 9 ft. beyond a vertical wall of a house. What is the height to which the ladder will reach ? 15. The area of a square field is 14 ac. 3 ro. 24 po. Find its side, true to the nearest link. 16. A window having 56 square panes of glass has a lighting area of 5378-24 sq. cm. Find the side of each pane. 17. A square lawn of 36 ft. side is surrounded by a gravel walk of a uniform width of 6 ft. What is the length of netting which will enclose the walk and lawn ? EXERCISES LXXIV. d. Graphic Methods of finding Square Root. 1. Verify by measurement the correctness of the following construction for finding square roots : EXERCISES 319 Measure off a distance OA equal to unity ; produce it to B so that OB represents the number N of which the root is required. Describe a semi-circle on OB as diameter. Draw AC at right angles to OB, then OC represents \/N. 2. Verify the construction in Ex. 1 in the following cases (i cm as unit) : N = 4 , N=9, N = i6, N = i-69, N = 3-6, N = 2-25, N=6-25. Show also that AC represents the square root of AB. This construction may be extended in its application, thus : 3. (a) Measure off OA = 3 in., OB = 5 in.; prove by measurement and calculation that OC represents the square root of the product of 3 and 5, i.e. of 1 5. (b) Extract graphically the square roots of (i) 10, (ii) 14, (iii) 30, (iv) 22-5. 4. Construction of square root curve. Mark off on squared paper two lines O.r, Ox, at right angles to one another. On the line xO mark off OL to the left, of length representing 4. Describe a number of circles, each with centre on LO.r, passing through L, cutting Oy say at M, and Ox at N. Draw MP, NP at right angles to Oy and Ox respectively. The various positions of P will lie on a curve such that PN represents the square root of ON. Use your curve to find (a) \/3, () v/22, (c) ^19-6, (d) ^2-89, each to one decimal place ; verify by calculation. EXERCISES LXXIV. e. 1. A boat starts at A and takes the following courses : Moves 3 miles to the East, but at the same time 4 miles to the N., then 6 6-3 then 4-5 3-8 State its final position ; also the distance between the initial position and the final. 320 A MODERN ARITHMETIC 2. In Ex. 1, state the final position, also the distance between the initial and final positions, when the courses were as follows : (a) (i) 2-21 m. to E., 1-08 m. to N. ; then (2) 3-26 m. to E., 0-8 1 m. to N. ; then (3) 1-85 m. to E., 0-21 m. to S. (b) (i) 6-1 m. to W., 3-2 m. to N.; then (2) 1-8 m. to E. ; then (3) 2-1 m. to E., 0-12 m. to S. (c) (i) 5-6 m. to E., 4-1 m. to N. ; then (2) 1-8 m. to E., 1-6 m. to S. ; then (3) 4-2 m. to E., i-S m. to N. 3. Complete the accompanying table, showing the relative lengths of the three sides of a right-angled triangle in which the hypothenuse has different inclinations : No. of Degrees in base angle. Horizontal Measure, Hypothenuse being i. Vertical Measure, Hypothenuse being i. (*) 10 0-98481 ? (*) 2O ? 0-34202 W 30 0-86603 ? (d) 40 0-76604 ? Approximations in Square Root (continued}. Suppose a number, say 104, to be very nearly an exact square; the first two figures in the square root are i and o, and 104 io 2 = 4. Two more figures are obtained by dividing 4 by twice 10, i.e. 0-20, and it is seen that whereas 104 differs from 100 by 4%, Vio4 = 10-2 approx. and differs from 10 by 2 %, and more generally if a number N ;/ 2 ,, x% X o/ This and the results on page 311 form the basis of a large number of examples of mental or approximate computations. APPROXIMATIONS IN SQUARE ROOT 321 EXAMPLE i. Evaluate (a) ^325, (b) ^3636. (a) \/375 = v'324 + * = \/i8 2 + i ^- approx. (b) ^3636 ; 3636 differs from 3600 by i %, = 60 increased by = 60-3 approx., while the error in the approx. will be J % of J %, *.*. T J^ %. EXAMPLE 2. A railway has a vertical rise of i in 250 horizontal. What is the length of rail corresponding to a rise of 4 ft. length 1 The length = \/4 2 + iooo 2 = 1000 + ^JJo- approx. = 1000-008 ft. EXERCISES LXXIV. f. Mental or Oral. 1. On a railway incline, the vertical rise is i ft. for every 100 ft. horizontal. What is the length of the incline for each 100 ft. measured horizontally ? 2. If in Ex. 1 the rise was i in 240, what was the length of the incline if the rise was 5 ft. ? 3. In an incline a vertical rise of 3 ft. occurs for every 150 ft. horizontal. Find an approximation to the length of the incline corresponding to a rise of 6 ft. 4. State the number of decimal places to which the following square roots can be found, the number under the root being approximate : (a) \/ioo-6.... (b} \/903 (c} ^400-04.... (d) \f 1 00000-008.... (e) ^625-35.... (/) \/4-44..., J.M.A. 3 22 A MODERN ARITHMETIC Square Boot (continued). Subjoined is a table of squares of the first 60 numbers. It may be employed to write down approximations to the square roots of many numbers. No. Square. No. Square. No. Square. No. Square. No. Square. No. Square. I I II 121 21 441 31 9 6l 41 1681 51 2601 2 4 12 144 22 484 32 J024 42 1764 52 2704 3 9 13 I6 9 23 529 33 1089 43 1849 53 2809 4 16 14 I 9 6 24 576 34 1156 44 1936 54 2916 5 25 15 225 25 625 35 1225 45 2025 55 3025 6 36 16 2 5 6 26 676 36 1296 46 2116 56 3136 7 49 17 289 27 729 37 1369 47 2209 57 3249 8 64 18 324 28 784 38 1444 48 2304 58 3364 9 81 19 361 29 30 8 4 I 39 I 5 2I 49 2401 59 3481 10 IOO 20 400 900 40 I6OO 50 2500 60 3600 EXAMPLE i. Evaluate 24-0-08 = 23-92, EXAMPLE 2. Evaluate ^2710. ^27 10 = ^2704 + 6 = 52+- = 52-06. 2x52 EXERCISES LXXIV. g. Use the above table to evaluate 1. ^66. 2. ^630. 3. v'i64o. 4. \/24i2. 5. ^1040. 6. ^3060. 7. ^236. 8. ^105. 9. ViQSo. 10. PROPORTION 323 52. PROPORTION (continued}. In questions of Simple Interest, several factors are involved. It has been seen (p. 278 et seq.) that the interest depends upon the (1) Principal '; and is, other things being the same, pro- portional to the principal ; (2) Time; and is, other things being the same, proportional to the time ; (3) Rate per cent.; and is, other things being the same, proportional to the rate. Consider now more general problems in proportion, in which several factors are involved. EXAMPLE i. If 18064 Ib. of biscuit formed the allowance to 2016 men for 35 days, what would be the equivalent allowance to 1050 men for 42 days? The factors involved are (1) Time; and the weight of biscuits would be proportional to the time if the number of men remained the same ; (2) Number of men; and the weight of biscuits would be proportional to the number of men, if the time remained the same. First, consider only the time. If 18064 Ib. be allowed to 2016 men for 35 days, then 18064 x Jf Ib. would be allowed 2016 men for 42 days. Next, consider the number of men. If 18064 x F be allowed 2016 men for 42 days, 18064 x | x ^Y would be allowed to 1050 men for 42 days, and the required allowance = 1 8064 x f | x lf Ib. = 1 1 290 Ib. EXAMPLE 2. If, in hiring goods^ the cost is determined as being proportional to the value of the goods and to the time of hire ; what would be the cost of hiring goods to the value 324 A MODERN ARITHMETIC " I 5 f r 5 da y s if the cost f "20. 5-r. for 12 days be i. los. o Here the hire is i. los. x $- x of *$ for 5 days if the cost of hiring goods to the value of 20. $s.for 12 days be i. los. od.t / 81 = 9^. $d. to the nearest penny. EXAMPLE 3. One locomotive A is known to travel 52 ft. for every 35 ft. that another locomotive B travels, when drawing their customary trains. If A completes a journey of 25-84 miles in 56 min.) how far will B go in i hr. 32 min.1 The factors involved in computing the distance are the speed and the time. The distance travelled by B speed of B time B travels ,. t , = -t- - x -T- r x distance A travels speed of A time A travels = 3^ x 9 x 25-84 miles 52 56 \ x 5 xkx 23 ., = - v- x 25-84 miles 13x^x^x8 115 x 25-84 .. = * - miles. 104 = 28-57 miles, to 2 decimal places. EXERCISES LXXV. 1. The weight of the load which can be supported by a wire is proportional to the area of the cross-section of the wire, and depends also on the nature of the substance of the wire. If a steel wire can support a load of 1-30 tons for every 0-75 ton supported by a copper wire of the same cross-section, what load can be EXERCISES 325 supported by a wire of copper of 0-2565 sq. in. cross-section, if a steel wire, section 0-1350 sq. in. can support a load of 7-02 tons? 2. The extension of a wire when stretched is proportional to the product of the stretching force and the length of the wire. If a wire 15-6 ft. long increases in length by 0-24 in. when stretched by a weight of 18 lb., what increase will be produced in the length of 12 ft. of the same wire when the load is 26 lb. weight ? 3. On a certain railway the passenger fares are computed as being proportional to the distance. If ,581. us. 6d. be received from 2705 passengers, each having travelled an average distance of 51-6 miles, what fares would be received from 1384 passengers, each travelling an average distance of 62-5 miles? 4. The cost of lighting a house by electric light is based on the number of lamps alight, and the time during which they are alight. In two houses the consumption is as follows : House A 25 sixteen-candle-power lamps, each alight on the average for 212 hrs. House B 17 sixteen-candle-power lamps, each alight on the average for 180 hrs. If the bill for electric light be ,5. los. $d. in house A, what is it in house B ? 5. In silvering articles by electrolysis, the amount of silver deposited is proportional to the strength of the electric current in amperes, if the time remains the same, and to the time if the current remains constant. If 1213-056 grams of silver are deposited by a current of 25 amperes in 12 hrs., how much silver will be deposited in 210 min. by a current of 36 amperes ? 6. With the data of Question 5, find in what time 352-6 grams of silver will be deposited by a current of 36 amperes. 7. A steam engine employed in pumping water from a coal mine is found to consume an amount of coal proportional to the product of the amount of water pumped up and the depth from which the water is raised. If 112 lb. of coal are burnt in pumping 9600 gallons of water from a depth of 600 ft., how much coal will be burnt when 12,000 gallons are raised from a depth of 750 ft. ? 326 A MODERN ARITHMETIC 8. The power (measured in horse-power) developed by a water wheel is proportional to the amount of water it receives per minute and to the depth of the fall. If a water wheel receiving 9000 gallons of water per minute, with a fall of 12 ft., develops a power of 12-5 horse-power; find the horse- power of a similar wheel with a fall of 10-8 ft. and a feed of 8500 gallons of water per minute. 9. When the wind blows directly upon a surface, such as a wall, the force exerted by the wind is proportional to (a) the area of the wall, (b} the square of the velocity of the wind. If wind blowing at 15 miles per hour exert a force equal to the weight of 270 Ib. on a wall 40 ft. long and 12 ft. high, what force would be exerted upon a wall 63 ft. long, 15 ft. high, by wind moving at the rate of 23 miles per hour ? (State your answer to the nearest Ib.) 10. The pitch of one screw A is 2-540 times as much as the pitch of another B; if the first, A, is turned through 260-5 revolutions in moving through a distance of 12-21 in., what distance will the second screw B move through in 452-5 revolutions ? State your answer to second decimal place. 11. If the weight of a square bar of iron 2-5 in. broad and 12 ft. long be 241-44 Ib., what is the weight of a square bar 4-75 in. broad and 7-5 ft. long ? 12. The average rain fall per month at Scarborough was 2-253 in., and it was calculated that, during a certain period, the fall of rain on a certain field of 6-345 ac. amounted to 25-36 tons. What would have been the total weight of rain received during the same period on a field of 5-281 ac. at Londonderry, where the average rain fall was 3-375 in. per month ? 13. Sheets of tin plate are generally sold in boxes, containing varying numbers of plates of different sizes. In one box there are 225 sheets, each 13! in. x 10 in., and of weight i cwt. i qr. 7 Ib. What will be the weight of the sheets in another box of 200 sheets, each 1 5 in. by 1 1 in., the thickness of the sheets being the same in each case ? MEASUREMENT OF ANGLES 327 53. MEASUREMENT OF ANGLES. Suppose OA and OB are two straight lines inclined to one another at any angle. With O as centre and any radius describe a circle cutting OA and OB at a and b respectively. If the length of the arc ab be measured (pp. 205, 206), and its length divided by the length of Oa the radius of the arc, then, AOB remaining the same, the result of the division will not depend on the particular length Oa chosen, and is known as the circular measure of the angle AOB. FIG. 47. O FIG. 48. Suppose a line OA rotated about O, into a position OA', in which the line points in a direction exactly opposite to OA. The angle turned through is two right angles ; a line such as OB making equal angles with OA and OA' would form a right angle with either OA or OA'. The circular measure of an angle of two right angles is the ratio of the length of the arc of a semicircle to the radius of the semicircle, and is denoted by the symbol ?r; its value is 3-14159... ; but the approximation - 2 T 2 - is often used. The divisions and subdivisions of the right angle are given in the following table : i right angle = 90 degrees (90). i degree = 60 minutes (60'). i minute = 60 seconds (60"). 32 8 A MODERN ARITHMETIC EXAMPLE i. Find the circular measure of an angle of 50 13' 12". 50 13' 12" = 50 13-2' since 60 seconds make one minute. 50 13-2' =50-22 since 60 minutes degree. But the circular measure of 180 is TT. i = 78^ = 0-279X3.14159... = 0-8775... EXAMPLE 2. Express in degrees, minutes and seconds an angle, the circular measure of which is 1-234. Since the angle of which the circular measure is TT has 180, .". the angle of which the circular measure is 1-234 has 180 x 1-234 7T = 180x1-234: approx. 3-HI59 222-12 3-I4I59 = 70-7030 = 70 42-18' (multiply the decimal part by 60 to reduce to minutes) = 70 42' n" approx. (multiply the decimal part by 60 to reduce to seconds). EXAMPLE 3. If an arc 0/6-2832 in. subtend an angle 0/72 at the centre of a circle of $-inch radius, what will be the length of the arc of a $-inch circle which subtends an angle of 20? The arc is proportional directly to the radius, and directly to the angle subtended at the centre. .'. length of arc = 6-2832 x |f x | in. approx. = 1-0472 in. approx. MEASUREMENT OF ANGLES 329 In the practical measurement of angles in drawing, a pro- tractor is usually employed. Figs. 49 and 50 show the method of marking off an angle, and also of measuring an angle. FIG. 49. Marking off a line OB making an angle of 40 with a given line OA. 40 O A FIG. 50. Measurement of an angle AOB (40). EXERCISES LXXVI. a. Graphic. 1. Describe a circle of 5 cm. radius ; on it mark off an arc also of 5 cm. ; measure the angle the arc subtends at the centre and check your result by describing circles of other radii and setting off arcs equal in length to the respective radii. 2. Construct a graph for the conversion of degrees into circular measure. 330 A MODERN ARITHMETIC (Set off degrees horizontally, scale i represented by a line I mm. in length ; at a distance of 18 cm., set off a vertical 15-7 cm. long representing TT on a scale where .5 cm. represents the unit of circular measure.) Use your graph to find the circular measure of the following angles ; verify by calculation : (<*) 30, 0)57, (') 60, (d) 72, (*) 112, (/) 153. 3. Describe any circle, and place within it chords, the lengths of which are the following multiples of the radius of the circle : (a) 1-732, (S) M76, (c) i, (d) 0-765, (*) 0-618. Find the angles subtended by the chords at the centre, and express them as sub-multiples (approx.) of 360. 4. Draw two straight lines CXr, Oy at right angles to one another ; along Ox measure any distance ON ; at N erect a perpendicular NP. Join OP. If NP be n times ON, find the value of the angle PON when n has the following values : (a) 2, (b) 3, (,-) 4, (d) i (*) , (/) J. 5. Suppose PON is a right-angled triangle, with the right angle at N ; measure the angle NOP when the rise NP is the following fractions of the distance OP : (a) o-i, (b) 0-2, (c) 0-3, (d) 0-4, (') o-S, (/) 0-6, (g) 0-7, (h) 0-8. EXERCISES LXXVI. b. 1. Calculate the circular measure of the following angles : (a) 60 20' 30', (b) 32 5' 17", ( c ) 19 12' 50", (d) 26 o' 10", (') '5 15' 15", (/) 34 i/ 5", Or) 12 3' 3", (h) 83 12' 18". . (Assume TT = 3- 1 4 1 59.) 2. Express in degrees, minutes, and seconds, the angles of which the circular measures are : (a) 0-23457, (b) 0-823214, (c) 0-98728, (d) 3-11891. EXERCISES 3. Complete the following table : (a) W w w Length of Arc in cm. Radius of Circle in cm. Angle subtended by Arc at Centre of Circle. 1-834 3-12 ? to nearest degree. ? 22-2 13 25-8 19 ? to nearest degree. p 10 r 85-0 ? 29 100 p 37 54. AREAS OF CIRCLES. EXERCISES LXXVII. Graphic. 1. Describe on squared paper circles of the following radii, and find their areas : (a) i cm., (&) 2 cm., (c) 3 cm., (d) 4 cm., (e) 5 cm., (/) 6 cm. Draw a graph showing the relation between the areas of the circles and the squares of their radii. Draw also a graph showing the relation between the circum- ferences of the circles and their diameters. Compare the two graphs and interpret your results. A B FIG. 51. FIG. 52. 2. Cut out a circular disc of paper or cardboard, divide it into triangles as in Fig. 51 and arrange as in Fig. 52. Note that the 332 A MODERN ARITHMETIC figure is practically a rectangle, and that one side of the rectangle is half the circumference of the circle, while the other side is equal to the radius of the circle. Hence deduce an expression for the area of a circle. 3. Cut out a sector of a circle of 6-inch radius, proceed as in Ex. 2, and deduce an expression for the area of the sector in terms of the radius and the arc. FIG. 53. 4. Suppose the space within a circle filled by circular con- centric pieces of cord as in Fig. 53. Suppose the cords all cut along a radius and straightened out as is the typical cord in Fig. 54. What figure will result? Deduce an expression for the area of the circle. It will now be clear that the area of a circle = (radius) x (circumference) *-::''''' = Tr? S - FIG. 54. 5. If a circle be divided up into a number of sectors all con- taining equal angles, it is clear that the areas of all the sectors are AREAS OF CIRCLES 333 equal, and therefore that the area of a sector containing n is equal to (area of the circle) = (^ 2 ). Describe sectors of a circle of 3-inch radius containing (a) 30, (b) 50. Calculate the areas of the sectors by means of the formula, and verify by the use of squared paper. Find the areas also by first measuring the lengths of the corresponding arcs and then multiplying by half the radius. 6. Fig. 55 represents graphically the relative distribution of land in Russia as (a) arable land, (b} meadows and pastures, (c) woods and forests and (d) infertile areas. Express the areas as percentages of the whole. FIG. 55- 7. Fig. 56 represents the division of the population of the United States according to occupation. If the in- dustrial class be 9-3 %, find the per- centage of the other classes shown. FIG. 56. 8. Construct four right-angled triangles, and in each case describe circles with the sides of the triangles as radii ; measure the areas of the circles and find the relation existing between the areas of the circles corresponding to each triangle. 334 A MODERN ARITHMETIC 9. The circles (a\ (b), (c) (Fig. 57) have areas proportional to the populations of China and its dependencies, the British posses- sions, and Russia with Finland respectively. If the population of FIG. 57 . China and its dependencies be 434 millions, find the other popula- tions represented. 10. Copy the given sector (Fig. 58) on squared paper, complete the circle, find the areas of the sector and the circle, calculate the angle of the sector and verify by using a protractor. FIG. 58. 55. THE PRISM AND CYLINDER. If a prism or cylinder be cut into a number of small prisms or cylinders of equal lengths by planes perpendicular to the length of axis, the pieces so formed will be clearly of equal volume and of equal lateral surface. Equal increases in the length of a prism or cylinder will therefore occasion equal in- creases in the volume and lateral surface ; and, since the volume and lateral surface both vanish when the solids are indefinitely THE PRISM AND CYLINDER 335 short, the volume of a cylinder or a prism is proportional to the length of the cylinder or prism, and so also is the lateral surface. But if the prism be i cm. long, it is clear that, corresponding to each square centimetre of the end, there is i cubic centimetre of volume. Hence, the volume is proportional to the area of the end of the prism, and the volume of a prism or cylinder = (area of end) (length), and .'. volume of cylinder = (7rr 2 )(/), where / is the length, r the radius of end. Again, if a piece of paper be wrapped tightly round a cylinder or prism, it will be evident that, in both cases, the lateral surface = (perimeter of base) (length), and therefore for a cylinder the curved surface = 27r/-/, where r = the radius of end and / the length of the cylinder. FIG. 59. FIG. 60. EXERCISES LXXVIII. a. Practical. [Apparatus: Sheets of metal or cardboard. Tins or beakers. Callipers. Scale. Coil of wire. Cylinders. Burette. Balance. Sq uared paper. Screw gauge. ] 1. Weigh the given irregularly shaped sheet of metal or card- board ; cut out a rectangular piece, measure its length and breadth and then weigh it. Calculate the area of the irregular piece, check your result by tracing its outline on squared paper and counting the number of squares. 2. From a piece of cardboard cut out a circular disc and also a square, the radius of the circle being equal to the side of the 336 A MODERN ARITHMETIC square. Weigh both and compare the area of the circular disc with the area of the square. 3. Pour water into a beaker or tin to a convenient depth ; run in more water from a burette until the depth is increased by i cm. ; note the volume run out from the burette. The volume in c.o will equal the area of the section of the beaker in sq. cm. Find the ratio of this area to the square of the diameter of the section of the beaker. 4. Weigh the given coil of wire of known density, measure its diameter by means of a screw gauge, calculate the length of the wire. 5. Measure the length and diameter of the given cylinder and calculate its volume ; check by finding the volume by displacement and also by finding the loss of weight in water. 6. Calibrate the given glass tube by introducing mercury into the bore and finding the length of the column at different places. 7. Calculate the volume of the given washer (a) by measure- ment, (b) by displacement. 8. You are given three pieces of wire of the same length and material ; compare their cross-sections (a) by weighing, (b) by use of a screw gauge. EXERCISES LXXVIII. b. Mental or Oral. In all the following exercises TT may be taken as -\ ? approximately. 1. What is the area of the end of a regular prism, having the volume 256 c.c. and length 32 cm.? 2. What is the internal volume of a hollow square prism of length 12 ft., outside edge of base i ft. and thickness i in.? 3. The curved surface of a cylinder is 44 sq. ft, the circum- ference 4 ft. What is the length ? 4. The curved surface of a cylinder is 44 sq. ft., the length i ft. What is the radius ? EXERCISES 337 5. What is the curved surface of a cylinder 3 ft. long, 7 ft. radius ? 6. Find approximately the volume of a cylinder 4 ft. long and 7 ft. radius. 7. Find approximately the volume of a cylinder 14 cm. long and 35 cm. radius. 8. Find approximately the length of a cylinder 7 in. radius and volume 308 cub. in. 9. Give an approximate expression for the volume of the sub- stance of a pipe 7 ft. long and i ft. internal radius, inch thick. 10. What weight of water could be contained in a cylindrical bucket i ft. deep and I ft. in diameter ? 11. How many Ib. does a steel cylinder weigh, 7 ft. long and i ft. radius, if i cub. ft. weighs 450 Ib. ? 12. What is the internal radius of a cylindrical tube, the internal volume of which is 198 c.c. and length 7 cm.? EXERCISES LXXVIII. c. 1. A solid cylinder is 15 cm. long and 8 cm. in diameter. What is its volume ? 2. A cylindrical boiler is 12 ft. long and 5-42 ft. in diameter. What is its contents ? 3. A well is 6-23 ft. in diameter. How many cub. ft. are in the well when it contains water to a depth of 50 ft.? 4. A pump has a piston 9-32 in. in diameter, the length of the stroke is 25 in. What volume of water does the pump lift per stroke ? 5. The internal surface of a pipe is 2704 sq. cm. and the contents 5822 c.c. What is the length of the pipe, also its internal diameter ? 6. What is the volume of the material of a pipe per foot length, if the inside diameter is 0-892 ft. and outside diameter 0-922 ft.? J.M.A. Y 338 A MODERN ARITHMETIC 7. What is the weight of 20 ft. of an iron pipe, 9 in. outside diameter and 1-2 in. thickness, if a cubic foot of iron weighs 450 lb.? 8. If a solid rod, 6-62 in. diameter, weighs 17-2098 lb., what is the weight of a rod of the same metal 4-8 in. diameter, but 2-25 times as long ? 9. Regarding an iron roller as a hollow cylinder, find the area it rolls per revolution, if its thickness be 1-8 in., its internal diameter 2 ft. 3 in., and its weight 1357 lb. (i cub. ft. of iron weighs 450 lb.) 10. Find the (i) volume and (ii) curved surfaces of the following cylinders : Diameter. Length. Diameter. Length. (a) 8-72 cm. 600 cm. (b) 32-2 cm. 125 cm. (c) 0-231 cm. 3 cm. (d} 51-3 cm. 12 cm. 11. A piece of iron wire, 30 metres long, weighs 8-553 8 T - m air and 1-1404 gr. less in water. What is its diameter? 12. In order to find the bore of a piece of thermometer tubing, mercury is let into it ; and it is found that 0-134 gram of mercury occupies a length of 40 cm. of the tube. What is the mean diameter of the bore ? (One c.c. of mercury weighs 13-6 gr. approx.) 13. Mercury is introduced into a piece of glass tubing ; at one place A it occupies a length of 8-19 cm. ; at B a length of 6-58 cm. ; and at C a length of 8-64 cm. Express the diameters at A and B in terms of that at C. 14. If a certain piece of wire weighs 63-6174 grams, and has a diameter of 0-300 cm., what is the diameter of a piece of the same length and material, but which weighs 44-1787 gr.? 56. SIMILAR FIGURES. EXERCISES LXXIX. a. Mental. 1. A, B, C, D are four cubes, A of i cm. edge, B of 2 cm. edge, C of 3 cm. edge, D of 4 cm. edge. Write down their volumes. EXERCISES 339 2. A, B, C, D are rectangular blocks of the following dimensions : Length. Breadth. Thickness. A 2 cm. I cm. i cm. B 4 cm. 2 cm. 2 cm. C 6 cm. 3 cm. 3 cm. D 8 cm. 4 cm. 4 cm. It will be noticed that all the blocks are of similar shape. Calculate their volumes, and mention anything you may notice about the result. 3. A, B, C, D are four blocks of the following dimensions : Length. Breadth. Thickness. A 4-25 cm. 1-3 cm. 0-8 cm. B 8-5 cm. 2-6 cm. 1-6 cm. C 17-0 cm. 5-2 cm. 3-2 cm. D 34 cm. 10-4 cm. 6-4 cm. Fill up the following table with reference to the blocks B, C, D Length of Block Breadth of Block Thickness of Block Volume of Block Length of A Breadth of A Thickness of A Volume of A B C D 4. A, B, C, D are rectangular blocks having dimensions repre- sented by the following : A, length / in., breadth b in., thickness, / in. B, 2.1 , 2b 2/ Find the multiples that the volumes of B, C, D are respectively of that of A ; and express the results in factor form. Illustrate the result with respect to block B by a figure. 340 A MODERN ARITHMETIC 5. The volume of a block of length 36-2 cm., breadth 19 cm., thickness ii-i cm., is 7634-58 c.c. Write down the volumes of the following blocks : (a) length 72-4 ; breadth 38 ; thickness 22-2 cm., (b) 362; 190; ii i cm., (0 18-1 ; 9-5 ; 5-55 cm. 6. A rectangular block of iron weighs 7 ton 4 cwt. What is the weight of another block of iron, half as long again as the first, half as broad again and half as thick again ? 7. A leaden bullet is 1-2 mm. in diameter and weighs ^ grams. What will be the weight of a bullet having the diameter 0-84 cm. ? 8. How many bullets of 2-4 mm. diameter could be obtained from a large ball of 172-8 cm. radius ? 9. A hollow shell has its outer diameter 1 2 in. and inner diameter 10 in. ; if, instead of being hollow, it had been solid throughout, the weight would have been 243 Ib. What was the weight of the actual shell ? Express your answer in factor form. EXERCISES LXXIX. b. Graphic. 1. Copy Fig. 6 1 in your squared paper exercise book. Mark any point O within the figure. Join O to each corner or salient point a, b, c, .... Produce Oa to A so that aA = O, produce Ob to B so that B = O... and so on. Join AB... and so obtain a figure AB... of the same shape as ab.... Find the ratio of (i) AB:^; BC :&:..., (ii) the area of ABC... to that of dbc.... 2. Repeat Ex. 1, but this time en- large the figure so that OA is 1-4 times Oa, OB is 1-4 times Qb and so on. 3. The given plan (Fig. 62) is drawn to such a scale that i acre is represented by i square inch ; (a) reduce it so that 4 acres are FIG. 61 EXERCISES now represented by I square inch, is represented by I square inch. enlarge it so that I rood - FIG. 62. 4. The areas of the circles marked A, B, C, D, E were employed on a chart to represent the population of the Colonial possessions and protectorates of England, France, the Netherlands, Bel- gium and Germany respec- tively. If the inhabitants of the Colonies, etc., of the British Empire number 400 x io 6 , what are the populations of the others ? 5. Draw a diagram show- ing, by the areas of circles, a comparison between the popu- lations of Rhodesia (580000), Nigeria (467280), Bechuana- land (38000), North Central Africa Protectorate (330780), Cape Colony (277000), East Africa (200000). FlG> 342 A MODERN ARITHMETIC 6. Draw a triangle, with sides twice those of Fig. 64. Suppose that your triangle represents the number (160) of pupils per 1000 of inhabitants in the Netherlands, draw triangles of the same shape to represent the number per 1000 in each of the following countries : Great Britain 150, Austria 140, Belgium 124, Italy 84, Roumania 50. FIG. 64. 57. INVERSE PROPORTION. Consider the following example in proportion : EXAMPLE. One train A travels at 30 miles per hour, another train B travels at 20 miles per hour. If A completes a journey in 3 hr. 20 min., how long will B take to complete the same journey ? If train A, travelling at 30 miles per hour, completes the journey in 200 minutes (3 hr. 20 min.), a train, travelling at i mile per hour, would complete the journey in 30 x 200 min. A train, travelling at 20 miles per hour, would complete the 30 x 200 journey in - - - minutes. 20 .'. train B would complete the journey in 300 min. or 5 hr. In this case .e. the time taken" by B = (time taken by A) speed of A speed of B' or, the times are inversely proportional to the speed. The following cases are further examples in which the proportion is inverse : (a) The base of a triangle is inversely proportional to its altitude, if the area remains the same. INVERSE PROPORTION 343 (b) The densities of bodies are inversely proportional to their volumes, if their masses be the same. (c) The time is inversely proportional to the principal if the interest and rate per cent, remain the same. A great number of problems involving direct and inverse proportion are best done by use of equations, following out the method adopted in the following typical examples : EXAMPLE i. Two bodies are of the same mass. In the first, the volume is 31-8 c.c., and the density is 8-82 gm. per c.c. What is the volume of the second if the density is 7-20 gm. per c.c. ? Mass of the first = mass of the second. Vol. of second x density of second = vol. of first x density of first. Vol. of second _ vol. of first x density of first density of second _3i-8x8-82 7-20 EXAMPLE 2. Two cylinders are of the same mass; the first A is 12-6 cm. long, 3-4 cm. in radius and each cubic centimetre of its substance weighs 6-42 grams. The second B is 16-2 cm. long and made of a substance, each centimetre of which weighs 11-2 grams. What is the radius of the second cylinder ? Here, square of radius of B x length of B x density of B = square of radius of A x length of A x density of A. .'. square of radius of B _ (square of radius of A) x (length of A) x (density of A) (length of B) x (density of B) ,. . _ , x /I2-6 x 6-42 radius of B = (3-4) x A / ^-. V n-2 x 16-2 On evaluating, the radius is found to be = 2-27 cm. 344 A MODERN ARITHMETIC und. NOTE. An approximate answer should always be found. This can frequently be done in several ways. Thus, roughly, radius = (3-4) x A/-o- = (3- Again, lies between 6 and 7 times 0-34. radius = 2-27 approx. These approximations, however, should be generally mental. EXERCISES LXXX. a. Oral or Mental. 1. Two rectangles are of the same area : the length of the first is 6 times the length of the second, and its breadth is 12 ft.; what is the breadth of the second ? 2. A rectangular leaden block, 1-6 ft. long, is melted down and recast into a block with a cross-section |- of its original amount : what is its new length ? 3. If a field could be ploughed by a certain number of men in 14 days, in what time could it be ploughed by 7 times the number of men ? 4. A certain field can be ploughed by a certain number of men in 6 days : in what time could a field of 3 times the extent be ploughed by twice the number of men ? 5. Two masses are of equal weight : one has a density 7, the other a density 2 ; the volume of the one is 12 cub. in.; what is tJr2 volume of the second ? EXERCISES 345 6. The bills for lighting two rooms were the same : in the first room there were 6 burners, each alight on the average 45 hours ; there were 5 burners in the second room : how long were they each alight on the average ? 7. The diameters of two wheels on a locomotive are 6 ft. and 3^ ft. respectively ; the first wheel makes 4200 revolutions in a certain journey : how many does the second wheel make ? 8. A is 50 % of B and 40 % of C : what percentage is B of C ? 9. A is 50 % of B, B is 40 % of C : what percentage is A of C ? 10. If 100 ft. of a wall 8 ft. high can be painted for ^10, what length of a 6 ft. wall could be painted for 12, ? 11. If a ream of paper 72 in. by 48 in. weighs 28 lb., what will be the weight of a ream of paper 24 in. by 16 in.? 12. The capital of a company A is |ths that of a company B : if company A declare a dividend of 10%, what dividend could be declared by B with twice the net profits ? 13. The simple interest on a certain sum of money was ,50 in 6 years : in what time would the interest on a sum half as large again reach the same amount, at the same rate of interest ? 14. Two equal sums of money are put out at simple interest, and both obtain the same return : in one case the rate per cent, per annum was 8 and the time 6 months ; in the second case the time was 9 months. What was the rate per cent. ? 15. The number of swings made by a simple pendulum in a given time varies inversely as the square root of its length : if a pendulum 9 ft. long makes 60 swings in a certain time, how many more would be made in the same time by a pendulum only 4 ft. long ? 16. If 10 men reap a field in 24 hours, in how many hours can 8 men reap it ? 17. 5 men can mow 4 acres in 5 hours : how long would it take 3 men to mow 8 acres ? 18. The carriage of 80 tons 4 cwt. for 50 miles is 11. 4.?.: what would be the carriage of 401 tons for 20 miles at the same rate per ton per mile ? 346 A MODERN ARITHMETIC Ions' 19. 8 gas burners in 32 hours use 1000 cub. ft. of gas : how long would the same quantity last with 6 burners using it ? 20. A tank full of water is emptied by a pipe having a sectional area of 14-4 sq. cm. in 1-6 hours. In what time could it be emptied by a pipe having a sectional area of 18 sq. cm. ? Examination of Quantities for Proportion. The following are examples in which tests are made with the object of finding out if one quantity is proportional, directly or inversely, to another. EXAMPLE i. In the table given below, discuss the connexion between the temperature at which water boils and the pressure of the air : Pressure in mm. .... 680 700 720 740 Temperature in degrees Centigrade - 96-92 97-72 98-49 99-26 Here both quantities increase together. If the temperature and the pressure were proportional, equal changes in pressure would mean equal changes in temperature. Plotting a table of differences, we find Difference in pressure 20 20 20 temperature 0-80 0-77 0-77. The last pair of differences might suggest however propor- tionality, but with an error in the first. ' evaluate ^6> Le ~ 7 ' 4 5' and and 720 98-49 : 700 i.e. 7-31. i.e. 7-16. These results are not equal, and the quantities are therefore not proportional. EXAMINATION OF QUANTITIES FOR PROPORTION 347 EXAMPLE 2. I 2 3 4 5 6 Diameter of wire in inches 0-500 O-4OO 0-300 0-040 O-O2O O-OIOO Resistance in ohms per 1000 ft. 0-040216 0-062837 0-11171 6-2837 25-I35 100-541 Note columns (2) and (4). It is seen that decreasing the diameter to y^th the original increases the resistance loo-fold. Test, therefore, if the resistance is inversely proportional tc the square of the diameter. Form a new row in which the number in row (2) above is multiplied by the square of the number in row (i). The result obtained is found to be I 2 3 4 5 6 0-010054 0-OI005392 0-0100539 0-0100539 0-0100540 0-0100541 From the equality of the products it is inferred that the resistance in ohms per 1000 ft. is inversely proportional to the square of the diameter in inches. The example may be treated graphically. Plot the re- ciprocal of the resistance against the square of the diameter ; if the quantities are proportional, the graph will be a straight line passing through the intersection of the reference lines. EXERCISES LXXX. b. Graphic. 1. The following table gives corresponding values of pressure and volume for a given quantity of air kept at constant tempera- ture. Draw a graph showing how the reciprocal of the volume changes with the pressure. What deduction can you make from the graph ? Pressure in gr.wt. \ per sq. cm. / 600 650 700 750 800 850 900 950 Volume in c.c. 37-26 34-40 31-94 29-81 27-94 26-30 24-84 23-50 34 8 A MODERN ARITHMETIC Find also from your graph (a) the pressure when the volume is 30 c.c., (b) the volume when the pressure is 825. 2. The following table gives the weight, length, breadth and thickness of a number of blocks of metal ; draw a graph connecting the weight with the volumes ; state which of the blocks are probably made of the same substance : (a) (b) (c) (d) (/) (g) Weight in gr. 1382 634 1268 1760 518-4 880 3857 Length in cm. 8 6 12 10 4-5 20 H Breadth in cm. 6 4 8 10 4 20 12 Thickness in cm. 4 3 6 8 4 i 2 (0 (/) (/) (*) Weight in gr. 1351 1910 862 1980 | 1555 720 Length in cm. 8-5 12 7 15 9 10 Breadth in cm. 8-5 12 7 15 8 10 Thickness in cm. 8-5 6 2 4 3 3. By the use of a suitable graph, show that the average number of yards per Ib. of bare copper wire is inversely proportional to the square of the diameter : Diameter in inches 0-128 0-064 0-048 0-036 0-028 Average no. of yds. per Ib. 6-7 26-9 47-9 84 140-5 4. The table below gives the average resistance in ohms per 1000 yds. of vertalin wire, and also the average number of yards per Ib. Show that the two sets of numbers are proportional : EXERCISES 349 Average number ot yds. per Ib. 3-085 3-84 4-88 6-37 8-65 Approx. resistance in ohms per 1000 yds. 39-65 48-9 62-1 80-9 no- 1 5. The following table gives the maximum current in amperes which can be obtained from accumulators having different numbers of plates, also the capacity in ampere hours. Show that the maximum discharge and the capacity are pro- portional. What would the quantities be for cells of 9 plates ? No. of plates - 3 5 7 ii 15 21 Max. discharge i-3 2-6 4-0 6-5 9-0 13 Ampere hours 9 18 27 45 63 90 amperes. 6. The following table gives the length of some of the longest rivers in the world, together with the areas of their basins. Draw graphs connecting the areas of the basins with (a) the lengths of the rivers, (b) the square of the length of the rivers. Make whatever inference you can from your graphs : Name of River. Length of River in hundreds of miles. Area of Basin in thousands of square miles. Volga - 23 579 Danube 18 315 Yang-tse Kiang 30 700 Yenisei - 29-5 970 Nile 37 II2O Congo - 28-8 1428 Amazon (proper) - 35 2833 La Plata 24 1200 7. The electric resistance of a piece of wire is proportional to its length, inversely proportional to the square of the section, and depends also on the nature of the substance of the wire. Find graphically which of the wires below are made of the same 350 A MODERN ARITHMETIC material by plotting the resistance against the product of the length and the inverse square of the diameter : (a) (6) (f) (d) i Resistance in ohms 570 19-5 131 64-6 II4O Length in yards 1000 500 IOOO IOOO 500 Diameter in inches 0-0360 O-2O 0-108 O-O22 O-Ol8o (A) Resistance in ohms 79-5 546 450 203 942 Length in yards 500 IOOO 2500 IOOO IOOO Diameter in inches 0-098 0-006 0-064 0-0124 0-028 8. The pressure exerted by a given gas is proportional to its absolute temperature directly, and to its volume inversely. Complete the following table, connecting the pressure, volume and temperature of a certain amount of gas : (a) (b) (c) (d} (e) (/) Pressure IOOO ? ? 900 600 ? Volume 50 80 90 IOO p 200 Temperature (abs.) 300 400 500 ? 700 800 9. The following is an extract from the catalogue of a dealer in scientific apparatus : EVAPORATING BASINS. Diameter - - 5$ 6^ 7^- 8 9^ inches, Capacity 18 30 45 70 90 ounces, Price - - 2/3 3 /- 4 /- 5 /- 5/6 each. Show graphically that the capacity is nearly proportional to the cube of the diameter, but that the price is not. What is (a) the capacity of an 8 in. basin, (b) the diameter of a basin of 50 oz. ? EXERCISES 351 10. The following are particulars respecting some small dynamos : Dimensions in inches. Weight in Ib. (a) 4 x 4 x 5 4 (b) 8^x5 x 5 8 (c) 9^x7 x; 19 (d) ii x8 xg 25 (*) 12^x9 x 9 40 Show that in four of these the weight is proportional to the product of the dimensions. EXERCISES LXXXI. Proportion. 1. The cost of 4 tons 16 cwt. of rails is ,18 : what will have to be paid for a truck load of rails weighing 12 tons when the cost per ton of the first to that of the second is as 3 : 4 ? 2. 15 men in 10 days of 9 hours dig a trench 30 yards long, 1-8 yards wide and 7-2 feet deep. In how many days of 8 hours each would 24 men dig a trench 40 yards long, 1-5 yards wide and 8 feet deep ? 3. The wages of 17 men for 12 days are ,56. 12s. : what should be the wages of 24 men doing similar work for 9 days ? 4. The provisions in a besieged town of 4500 people are sufficient to last 54 days, but after 21 days 1000 more people take refuge in it. How long will the provisions now last, at the same rate as before ? 5. In supplying coal gas from a gas works, the number of cub. ft. of gas, forced per hour through a given length of pipe, is proportional to the square root of the pressure of the gas. If 8873 cub. ft. of gas can be supplied in 60 min. by a pipe 5000 yds. long and 9 in. in diameter, when the pressure is estimated as 3-000 in. by a water gauge, what time would be taken in delivering 21,635 CUD - ft- when the pressure is 2-000 in. by the water gauge? 352 A MODERN ARITHMETIC 6. In what time will 323-2 gr. of silver be deposited by a current of 33 ampkres, if 1213-056 gr. be deposited in 12 hrs. by a current of 36 amperes ? 7. Two metal plates of the same substance are weighed and partly measured, with the following results : First metal plate, 18-03 cm. long, 15-06 cm. broad, thickness 1-12 cm., weight 3060 grams. Second metal plate, 9-24 cm. long, 7-12 cm. broad, thickness not observed, weight 1530 grams. What is the thickness of the second plate ? State your answer as accurately as the data allow. 8. A contractor undertakes to complete a contract in 42 weeks, by employing 64 men. At the end of 28 weeks he finds that only one half of the work is done. How many more men must be put on so as just to complete the contract in the specified time ? 9. If 2000 soldiers have provisions for 95 days, how many men must be sent away in order to make the food last 100 days ? 10. A ship with a crew of 32 sailors has food enough for 84 days, but at the end of 42 days 16 sailors are picked up : how long will the food now last, supposing the rations to continue unchanged ? 11. If 12 cub. ft. of coal gas can be obtained from Cannel coal for every 10 cub. ft. contained from the same weight of Newcastle coal, how many tons of Cannel coal will be required to produce 1,050,000 cub. ft. of coal gas, if 15 tons of Newcastle coal are required to produce 135,000 cub. ft. of the gas? 12. Since the earth is nearly spherical, the top of the mast of a ship at sea can be seen before the lower part, when viewed from a point on the sea level. The distance at which the top of the mast can just be seen is proportional to the square root of the height of the mast. If a mast 54 ft. high can be seen at a distance of 9 miles, find (a) the distance at which a mast 16-67 ft- high can be seen ; (b) the height of the mast of a ship which can just be seen at a distance of 6 miles. EXERCISES 353 13. When 1800 men have provisions for 114 days, they are joined by 400 men. How long will the provisions now last? 14. Assuming that the cost of carriage is proportional to the weight, and also to the distance carried : (a) What weight could be carried 2 miles 7 fur. at the same cost as 3 cwt. .1 qr. 27 Ib. could be carried 48 miles 7 fur.? (b) How many miles could 11 tons be carried for ,5. los. if 3 tons be carried 54 miles at a cost of ,2. os. 6d.~? (c) If the carriage of 3 cwt. 3 qr. 14 Ib. for 104^ miles cost \. 14-y. lort 7 ., what should be charged for the carriage of 5 tons for a distance of 93 miles? 15. 8 men can finish a piece of work in 75 days by working 10 hours a day. How many more men must be put on in order to finish the work in 40 days without increasing the number of hours ? 16. If a lime kiln can burn on an average 2688 cub. ft. of lime in 12 days, find the least number of kilns to be employed if it be required to burn 97342 cub. ft. of lime in not more than 7 days. 17. On account of capillary attraction, the rise of water up a glass tube of narrow bore is inversely proportional to the diameter of the bore. The rise is 3-14 cm. when the diameter is 0-0513 cm.; what is (a) the rise when the diameter is 0-0273 cm -> (&) the diameter if the rise is 4-27 cm. 18. The pressure required to compress a given quantity of air kept at constant temperature to a given volume varies inversely as the volume. If it requires a pressure of 786-4 grams weight per square centimetre when the volume is 38-90 c.c., find (a) the pressure when the volume is 25-89 c.c., (b) the volume when the pressure is 3 kilograms per square centimetre. 19. If 4 Ib. of bread cost ^\d. when wheat is 44.9. a bushel, what will 6 Ib. of bread cost when wheat is 48^. a quarter, assuming the other factors in the cost to vary according to the price of the wheat? 20. If it cost as much to feed 3 men as 4 boys ; and if for 3 boys the cost is 19.?. *2\d. a week, how much a week will it cost for 51 men? J.M.A. z 354 A MODERN ARITHMETIC 21. When the cost of food for 84 men for 95 days amounts to ,598. ioj., what will be the cost of feeding 1 10 men for 33 days ? 22. In India, where artificial irrigation is generally required, a field of rice at Mysore, 25-3 ac. in extent, requires the water from a source supplying 120-6 cub. ft. per sec.; at another field where the soil is different 35 cub. ft. are required for every 30 cub. ft. at Mysore. What flow per sec. will be required to irrigate the latter field, if its extent be 36-3 ac. ? (State answer to nearest tenth of a cubic foot.) 23. The pressure a cylindrical boiler can sustain without bursting is directly proportional to the thickness of the boiler plates and inversely proportional to the diameter of the boiler. A boiler of 3-5 ft. diameter, with plates of in. thickness, can withstand a pressure of 573 Ib. to the sq. inch : (a) What is the diameter of a cylindrical boiler, plates | in. thickness, which can just withstand a pressure of 418 Ib. to the so. inch? (b) What pressure can a boiler, plates f in., sustain if its diameter is 2 ft.? (State the answer to the nearest pound per sq. in.) 24. A ship's crew of 18 hands had food enough on Jan. ist, 1905, to last 48 days, but on that day they picked up 6 shipwrecked sailors : how long will the food last ? 25. When wheat is 15.?. a bushel, 8 men can be fed for 12 days at a certain cost : for how many days can 6 men be fed for the same cost when wheat is 12s. a bushel? 26. A contractor had engaged to make a railway 58-5 miles long in 40 weeks, and began by employing 2160 men. At the end of 13 weeks, 19-5 miles were completed. How many men can be dismissed if he only just wishes to complete the work at the end of the 40 weeks ? 27. The following table gives the velocity acquired by a body in falling from rest through different heights : Fall I 4 25 100 400 feet Velocity | 8 16 40 80 160 ft. per second EXERCISES 355 What is the connexion between the fall and the velocity acquired ? Find also (a) the velocity acquired during a fall of 375 ft. ; (b) the fall necessary in order that a velocity of 98 ft. per second may be acquired. 28. The speed, or velocity, with which waves move on the surface of the deep sea is proportional to the square root of the length of the wave. If the velocity of a wave 50 ft. in length be 12*85 ft- per sec., what will be the velocity of a wave 120 ft. in length ? (State the answer true to o-oi of a ft. per sec.) 29. With the assumption made as in Ex. 6, find the time in which a current of 16-32 amperes would deposit 10-83 grams of silver. 30. If a rod of larch can support a load of 10-2 x io 2 Ib. weight per sq. in., what can it support in kilog. per sq. cm.? 31. The corrosion of iron and steel on account of 'rusting' is proportional to the surface. When cast iron is placed in clear sea water the amount of corrosion is 0-0635 Ib. per year for every square foot of iron exposed, but in fresh water the amount is 0-0113 Ib. per year for every square foot: find the amount of corrosion taking place in 4 years in a pipe, 12-62 cm. radius, lying in sea water, if a pipe of the same length, but io cm. diameter, lying in fresh water corrodes to the extent of 30-3 kilograms in 3 years. 32. Compare the thickness of slate, known as doubles, size 13 in. x6 in., and looo of which weigh 15 cwt., with that of slate known as Countesses, size 20 in. x io in., 200 of which weigh 8 cwt. 33. In a non-manufacturing town the consumption of water is 17-5 gallons of water per person per day ; the reservoir contains 200 days' supply, and receives the water from an area of 10,000 acres. What would be the number of days which a reservoir would supply a manufacturing town of the same population, but in which each person consumed 24 gallons daily, and in which the reservoir received the water from an area of 18,000 acres. (Assume the same conditions in both towns with respect to rain- fall, evaporation, and so on.) 356 A MODERN ARITHMETIC 34. If 12-3 sq. yd. of a fine sand filter can filter 1000 gallons of water in 24 hourc, how many sq. yd. of a similar filter would be necessary to filter 13,500 gallons of water in 15 hours ? 35. Two pieces of copper wire are ot the same length ; one weighs 0-632 gram and its section is 1-12 mm. in diameter; the second weighs 0-835 gram: what is its diameter? (State your answer as accurately as the data permit.) 36. The following table gives the total load that a horse, moving at different speeds, can draw in a canal boat, along a canal : Load 520 243 153 1 02 52 30 19 13 9 6-5 tons Speed 2-5 3 3-5 4 5 6 7 8 9 lOmls. per hour Show that the load is neither inversely proportional to the speed nor inversely proportional to the square of the speed. 37. The length ot i on the meridian is 68-70 miles near latitude o and 59-23 mis. near latitude 60. If a certain length, measured along the meridian near latitude 60, be 0-08764 of a degree, what fraction of a degree would the same length be near the equator ? 38. A commences business with a capital of ,1875 ' at trie er >d of 5 months B joins him with a capital of ,1500, and at the end of 4 months more C enters with ,1400. The profit at the year's end was 1116: how should this be divided between A, B and C? 39. Four people enter into a business, A with a capital of ,868, B with ,574, C with ,794 and D with ,764. A is to receive a fixed salary of .120 from the profits, and the remainder is to be divided among the four in the proportion of their subscribed capital. What amount does each receive from a profit of ,270 on the year's working ? 40. Two persons rent a grazing marsh for ,10. 12s. a year. One puts in 420 sheep, but at the end of 7 months puts in 96 more for the remaining months of the year ; the second puts in 760 sheep for 6 months, and then withdraws 320 for the rest of the year. How should the rent be apportioned ? EXERCISES 357 41. Two workmen are engaged on a piece of work for which they are to receive 2 guineas. One workman works for 3 days of 9 hours a day, and the other 3 days of 8 hours a day ; but the second does as much in 2 hours as the first does in 3 hours. How should the amount be divided between the two workmen ? 42. The boarding expenses of a party of 14 persons for 3 weeks were ^63. What would be the expenses of a party of 8 persons for 5 weeks ? 43. The time taken in emptying a canal lock by a sluice is proportional to the square root of the head of water, and inversely proportional to the area of the sluice. If, in a certain lock, the time taken is 2 min. 20 sec., with a certain sluice and a head of 15 ft, find the time taken in a similar lock, with a sluice the area of which is half as much again, but in which the head is only 14 ft. (State the time to the nearest second.) 44. Two reels of copper wire are weighed: the first weighs 1-309 Ib. including the reel, the wire is No. 13 wire gauge (diameter 0092 in.); the second weighs 6-66 Ib. and wire is No. 30 (diameter 0-0124 in.)- If the length of wire on the first is known to be 46-52 ft., what is the length of wire on the second, the reels in each case weigh 0-13 Ib. ? 45. If a lath be supported at its ends, and loaded at the centre, it bends, and its centre is lowered by an amount pro- portional to the load, inversely proportional to the cube of the thickness, inversely proportional to the breadth, and directly proportional to the cube of the length of the lath. If a load of 2053 grams produce a lowering of 5-21 mm. at the centre of a lath 100 cm. long and 0-420 cm. thick, find (a) the lowering at the centre when a lath of the same wood and breadth is used, but the length of which is 122-0 cm. and the thickness 0-260 cm., the load being 1600 grams ; (b) the load necessary to produce a deflexion of 1-20 cm. in a lath similar to that used in (a\ but 96-0 cm. in length instead of 122-0 cm. State your answer true to the accuracy possible from the given data. 358 A MODERN ARITHMETIC tional 46. The strength of a beam or girder is inversely proportiona to its length, directly proportional to its width, and directly pro- portional to the square of its depth. A wooden beam, of rectangular cross-section, is 4 ft. long, 2 in. wide, 3 in. deep, and it rests upon supports at its ends. The breaking load is 2000 Ib. placed at its centre. What would be the breaking load of the same beam (<z)when the supports are brought 2 in. each from the ends and (b) when the beam is placed so as to be 3 in. in width and 2 in. deep ? 47. A standard beam of wood, i in. wide, I in. deep and 12 in. long, can safely carry 420 Ib. at its centre. What must be the depth of a beam of the same wood, -2\ in. wide and 5 ft. long, to carry safely a load of 1-5 ton ? (See Ex. 46.) 48. If wind blowing at 15 mis. per hour exerts a force equal to the weight of 270 Ib. on a wall 40 ft. long and 12 ft. high, find the velocity of the wind if the force exerted on a wall 54 ft. long and 1 6 ft. high be 1728 Ib., assuming that the force exerted is pro- portional to the square of the velocity. 49. When waves travel on the free surface of shallow water (the length of the wave being much greater than the depth of the water) the velocity of the wave is proportional to the square root of the depth. If a long wave take 4-5 sec. in describing a distance of 36 ft. when the depth is 2 ft. : (a) How long will a wave take in describing a distance of 125 ft. when the depth is 5-12 ft.? (State the answer true to 2 significant figures.) (V) What would be the depth if the wave moved with a velocity 4-62 metres per sec.? 58. AREAS OF RECTILINEAR FIGURES. EXAMPLE i. To find the area of a trapezium ABCD. Suppose in Fig. 65 AB and DC are at right angles to AD, and that the lengths of AB, DC, and AD are known. AREAS OF RECTILINEAR FIGURES 359 If AB=_y 1 , DC=j^ 2 ' an d if A is at a O distance x l from a certain point O along DA, while D is at a distance x 2 ; then B Draw BB', CC' both parallel to AD meeting DC produced and AB at B' and C' respectively. Then ADB'B is as much greater than ABCD as ABCD is greater than ADCC'; and therefore the area ABCD is half the sum of the areas of AC'CD and ABB'D; .'. area ABCD = x AD. A FIG. 65. the area ABCD EXAMPLE 2. To find the area of a small rectilinear plot from field-book readings taken with reference to a single base line. Suppose ABCDE (Fig. 66) represents the plot, Qabecd the base line, and that CASE I. All the points AB... are on the same side of the base line. Let the field-book entry be as follows : y* x FIG. 66. Alter the order of the readings to correspond to the case where the order is that in which the corners would be met by 360 A MODERN ARITHMETIC a man going right round the field ; and, say, starting at A, the entries would now be X M X n X A o in which the order is ABCDEA. The area of ABODE = area of ABCD^- area - dDEe - (x o - X B ) (y c +y B } + (x n - x c ) (J D +y c ] - (x E -x A )(y s +y A )} = %{(XB~ X A) (y B +?A) + ( x c ~ X B) (jc +7*) + (X D - X c ) (y D +y c ] + (x E -x D )(y E +y D ) + (x A -x E )(y A +y E )} (i) CASE II. The points observed are on different sides of the base line. Suppose in the preceding ^__^ ,C case that the field-book ^p"""""" \ entry had been with refer- o' a' / b' e' c' \d ence to a line like Q'a'b'e'c'd' ^ / \ (Fig. 67), parallel to Qabecd, \ and at a distance r from it, \ \ / then, if the new field-book \ ^ readings are 'V X jf E x c a X E y' E y ' B X B O b e c d X A y' A FIG - 6 7- O it can be easily seen that y B = y c = but that A = AREAS OF RECTILINEAR FIGURES If these values be substituted in formula (i), it will be noticed that the terms in r all cancel, and therefore that the expression is just of the same form as in (i), only the distances from the base line are counted as negative if the points are to the right of the base line, and positive if to the left of the base line. EXAMPLE 3. Find the area of a plot from the following field- book readings : 100 60 30 300 2 5 200 150 120 O 20 80 Here the right order is clearly obtained by first taking the points on the left going up the base line, and then the points on the right coming back along the base line; and the area expressed in sq. links = \ {(200- 1 20) (60 + 30) + (250 - 2oo)(ioo + 6o) + (300 - 25o)( - 20 + 100) + (150 - 3oo)( - So - 20) + (120- I50)(30-80)} = I {80 x 90 + 50 x 160 + 50 x 80 + 150 x ioo 4- 30 x 50} = 1(7200 + 8000 + 4000 + 15000 + 1500} = 4(35700}. The area is therefore 0-1785 acre. The expression J {(*2 - *l) (ft +^l) + (*8 - *2> (ft +-^2) . . . fo - *) 0^ + y n ) } may be written also in a form which is sometimes more con- venient for calculation. Thus : 362 A MODERN ARITHMETIC in this case the expression can be set down thus, and the several components { x \y<i~y\ x i are found by cross multiplication. y* EXAMPLE 4. Find the area of Fig. 68 from the data given : Oa=io, aA=io; 0^=14, #B = 2'0; Or=3o, ^rC = 2o; 0^=23, dD=io; Qe=2o; <?E = 5 units of length. FIG. 68. Area=-J , (10x20-10x14)= 60 -320 the cross multiplication. .. - 160 10 10 14 20 30 20 23 10 20 5 10 10 .'. area = J{ -60 + 320 + 160 + 85,- 150} . *=*r = 177-5 umts f area - ,;^) -' .products are shown. to 'the right. AREAS OF RECTILINEAR FIGURES Using the other form for the expression for the area, area = {(14- io)(2o+ io) + (3o- 14) (20 + 20) + ( 2 3 - 3)( 10 + 20 ) + (20 ~ 23X5 + I + (10-20X10 + 5)} = J {120 + 640 -2 10 -45- 150} = i(355) = 177-5 units of area. 363 EXERCISES LXXXII. Find the areas of the plots given by the following field-book readings (expressed in links) : 1. 100 500 80 400 60 200 3. 20 1900 10 1600 30 1 200 5. 600 5 5 400 20 300 200 IO 10 100 7. 50 250 160 200 40 150 80 IOO 10 80 2. 80 60 6. 8. 20 10 TO 10 500 400 2OO IQOO 1600 200 i6o 150 120 80 50OO 4OOO 3000 2000 1000 | IOO 20 10 2O 20 10 364 A MODERN ARITHMETIC 9. Find the areas of the Figs. 69, (a) to (/), by choosing base lines and marking off readings. Check your results by (i) varying the base line, (ii) by the use of squared paper.* FIG. 69. When the areas are not bounded by strictly rectilineal figures, the method can only give approximate results. 10. Find readings after the manner of field-book readings for the more salient points of Figs. 70, (a) to (c) ; calculate the areas from the readings and check by the use of squared paper. FIG. 70. 11. At equal distances apart' draw a number (seven at least) of parallel ordinates across Fig. 71 (a), join the ends of the ordinates, calculate the area of the rectilineal figure so formed, note that this area must be nearly equal to that of the given figure and hence get an approximation to the area of the figure. * In Exx. 9, 10, 11, and 12 it may be desirable for the student to draw larger diagrams for himself. EXERCISES 365 12. Obtain an approximation to the area of Fig. 71 (b) after the method suggested in Ex. 11. 13. Find the area of the triangle, of which the angular points are (o, o), (XB, ya\ (b, o). Prove from your result that the area is one half the pro- duct of the base into the altitude. FIG. 71. MISCELLANEOUS PROBLEMS LXXXIII. 1. The minute-hand of a clock passes over 60 divisions on the face of the clock for every 5 divisions that the hour-hand passes over, i.e. gains 55 divisions in one hour. The hands are together at 12 o'clock : find when next the minute-hand will be 15 divisions ahead of the hour-hand, i.e. when the minute-hand has gained 15 divisions. 2. Find when the hands of a clock are pointing in the same direction between (a) 5 o'clock and 6 o'clock (i.e. has gained 25 divisions since 5 o'clock), (b} 3 o'clock and 4 o'clock, (c) 10 o'clock and 1 1 o'clock. 3. Find when the hands of a clock point in opposite directions between the hours of (a) 2 and 3, (b) 4 and 5, (c) 9 and 10. 4. If a train, 80 yd. in length, and travelling at the rate of 60 miles an hour, overtakes another train, 70 yd. in length, moving at 35 miles an hour, it must do so at the rate of 25 miles an hour. Find how long such a train would take in passing the second completely, i.e. in gaining (80 + 70) yards on the second. 5. A train, 86 metres long, and moving at 40 kilometres an hour, overtakes another train, 78 metres in length, moving at 26 kilometres an hour : how long does it take completely to pass it ? 366 A MODERN ARITHMETIC 6. If the trains in Question 5 had been moving in opposite directions, what time would have been taken in passing one another? 7. If the waste-pipe can empty a bath in 5 min., and the hot- water pipe can fill it in 15 min., and the cold-water pipe in 10 min., calculate how much water enters or leaves the bath in i min. when (a) The hot-water pipe alone is open. (b) The hot and cold-water pipes are open. (c) The three pipes are open. Find also (d) the time taken in emptying the bath if all the pipes are open. 8. Two pipes are such that a cistern fed by them could be filled by each separately in 7 min. and 1 1 min. respectively : in what time would the cistern be filled by the two taps together ? 9. A can give B 40 yd. start in a quarter-mile race ; B can give C 40 yd. start in a similar race. If A can run the quarter- mile in 58 sec., how long would C take to run it, assuming B and C to run the full quarter at the same rate as they ran the portions mentioned ? 10. A can just give B 10 yd. start in a 100 yd. race ; B can just give C 15 yd. start : what is the utmost start A can give C? 11. One person starts from a town A to meet another person coming from a town B 12 miles off. If the first person walks at 3 miles an hour, the second at 4 miles an hour, and if they start at the same time (5 p.m.), at what rate do they approach one another, and when and where will they meet ? 12. The period of revolution of the earth round the sun is 365, days, that of Mars (in the same direction) is 687 days. Show that, if the motion was uniform, the apparent year of Mars with regard to the earth would be 770 days (i.e. show that this would be the time in which the earth would gain one revolution with respect to Mars, so that the two planets would again be in the same straight line with the sun). 13. The apparent year of Venus as seen from the earth is 584 days : show that the actual year (on the same supposition as in the preceding question) is about 225 days. MISCELLANEOUS PROBLEMS 367 14. Find also the apparent year of Venus, if observed from Mars. 15. A person mixes or blends two kinds of tea ; one kind is worth is. ^d. per lb., the other worth 2s. per Ib. ; the mixture is worth i s. *jd. per lb. ; so that for every lb. of tea of the first kind there is a gain of 3^., but for every lb. of the second kind there is a loss of ^d. Show that 5 lb. of the first kind must be mixed with every 3 lb. of the second kind. Find the amount of tea of the second kind to be mixed with I ton 5 cwt. of the first kind, and what gain per cent, on the cost price would result if the blend be sold at 2. los. per quarter. 16. How would you blend two teas worth respectively is. $\d. and 2s. id. per lb. so that the mixture is worth is. 7\d. per lb.? 17. In what proportion must a merchant mix one kind of tea at 3>r. with another at is. >d. in order that by selling the mixture at 2s. &d. he may make a profit of 25 per cent.? 18. A trairi leaves London for Brighton at 9 a.m., travelling at a uniform rate of 15 miles an hour. An express train leaves Brighton for London at ro a.m. and travels at a rate of 40 miles an hour. How far apart are the trains when the train for London starts, at what rate do they approach each other, at what time will they pass each other and at what distance from London, the distance from London to Brighton being 50 miles ? 19. A hare sees a hound 176 yards away from her, and scuds off in the opposite direction at a speed of 12 miles an hour; thirty seconds later the hound perceives her and gives chase at a speed of 1 8 miles an hour. How soon will he overtake the hare, and at what distance from the spot whence the hare took flight ? 20. A starts from London for Epsom, distant 14 miles, walking at the rate of 3 miles an hour. B starts from London on the same road if hours later, driving 8 miles an. hour. At what distance from Epsom will B overtake A .? 21. The first of a series of cogged wheels, working into each other in a straight line, has a certain number of teeth ; the number of teeth in the second is to that of the first as 6 : 7 ; of the third to the second as 5:6; and of the fourth to the second as 2 .: 3. If 3 68 A MODERN ARITHMETIC the wheels are set in motion, how many revolutions must each wheel make before they are simultaneously in their original positions ? 22. By selling goods for a certain sum a man gains 5 per cent, on the cost price. If he had sold them for 3^. more he would have gained 6 per cent. Find their cost price. 23. A cistern which can be filled by one tap in 28 hours is emptied by four others. The first of these alone would empty it in 10 hours, the second in 12 hours, the third in 15 hours and the fourth in 21 hours. Supposing the cistern to be full, in what time would it be emptied if all the five taps were set running together? 24. There is a well containing 750 gal. of water; two pumps raising 20 and 30 gal. per minute respectively are employed to empty it, while it is constantly supplied by a spring which can refill it in half an hour. The two pumps work together for 15 minutes, when that of larger capacity ceases work for 10 minutes ; the two pumps then work together until the well is empty. How long will each pump have been employed ? 25. Two boats start for a race at 3 o'clock. The race is over at 6| minutes after 3, the losing boat being 40 yards behind at the finish : at 4 minutes past 3 this boat was 700 yards from the winning post. Find the length of the course, and the speed (supposed uniform) of each boat in miles per hour. 26. A tricycle, going at the rate of 5 miles an hour, passes a milestone, and 14 minutes afterwards a bicycle, going in the same direction at the rate of 12 miles an hour, passes the same milestone : find when and where the bicycle will overtake the tricycle. 27. Two clocks are together at 12 o'clock: one loses 7 sec. and the other gains 8 sec. in 12 hours: when will the faster be half-an-hour before the other, and what o'clock will it then show ? 28. Two watches are together at 12 o'clock noon. One gains 75 sec. per hour and the other loses 45 sec. When will they be together again at 12 noon? 29. If A, B and C could reap a field in 18 days ; B, C and D in 20 days ; C, D and A in 24 days ; and D, A and B in 27 days : in what time would it be reaped by them altogether ? MISCELLANEOUS PROBLEMS 369 30. If 4 men can do a piece of work in 7 days, which 5 women can do in 8 days, or 7 boys can do in 10 days : how long will it take 3 men, 2 women and 3 boys to do the same if they work together ? 31. A can do a piece of work in 1 1 days, B in 20 days and C in 55 days : how soon can the work be done if A is assisted by B and C on alternate days ? 32. If 6 women and 3 boys weed -fe of a field in 4 days, how long will it take 4 women and i boy to weed the rest of it, the work of 4 women being equal to that of 5 boys ? 33. A certain field could be reaped by 7 men in a certain time, and 5 boys could do as much as 2 men. Find how many boys would be required, in addition to 30 men, for the reaping of a field of twice the size, in a third part of the time. 34. An astronomical clock has its dial divided into 24 divisions instead of 12, and the small hand gees round once in 24 hours, the large hand going round once every hour. The 24th hour is noon. Find when the hands are at right angles to one another between 24 and i, and find also the interval between two successive meetings of the hands. 35. Strong spirit is mixed with inferior spirit, valued at 55-. per gallon, in the proportion of 6 to i. The mixture is then worth gs. the gallon. Find the value of a gallon of the strong spirit. 36. A certain turnip-field can be hoed by 10 women in 4 days, or by 6 boys in 10 days, or by 2 men in 12 days. One man, three boys and three women are employed : what is the total cost of hoeing the field, a man's daily pay being 2s. 8</., a woman's is. 8^., and that of a boy is. ? 37. A milkman pays is. id. per gallon for his milk; he adds' water and sells the mixture at id. per pint, thereby making alto- gether 40 per cent, profit. Calculate the proportion of water to milk his customers receive. 38. A certain piece of work was done by A, B and C, all working together, with the exception that B left off half an hour before the completion of the work, A alone could have done the whole in J.M.A. 2 A 370 A MODERN ARITHMETIC yi hours, B in 8J hours, C in 10 hours. In what time was the work done ? 39. I row against a stream flowing i miles an hour to a certain point, and then turn back, stopping two miles short of the place whence I originally started. If the whole time occupied in rowing be 2 hr. 10 min., and my uniform speed in still water be 4^ miles an hour, find how far upstream I went. 40. A and B start from the same point to run in opposite direc- tions round a circular race-course, 9755 feet in circumference, A not starting until B has run 105 feet. They pass each other when A has run 4850 feet. Which will first come round again to the starting point (their speeds being uniform throughout) and what distance will they then be apart ? 41. A and B start at the same time from London to Blisworth. A walking 4 miles an hour, B riding 9 miles an hour. B reaches Blisworth in 4 hours, and immediately rides back to London After 3 hours' rest he starts again for Blisworth at the same rate How far from London will he overtake A, who has in the meantime rested for 6 hours ? 42. With what two quantities of spirits, that cost me 14^. 6d. am 17-r. 6d. a gallon respectively, did I make a blend of 12 gallons, i by selling the mixture at i6s. $d. a gallon I should have incurred loss of i\ per cent. ? 43. A man rode a bicycle from A to B, 54 miles, at an averag< rate of 8 miles an hour ; another man started from A on horsebacl half an hour after the bicyclist and arrived at B 15 minutes befor him. Find the ratio of their speeds. REVISION EXERCISES LXXXIV. Miscellaneous. (Mainly from examination papers.) A 1. Divide .720. 1 1 j. $d. by 2. 4j. sf //. 2. A man bought 10 tons 3 cwt. of sugar at ^18 a ton and sol I it at i\d. a pound. What profit did he make ? REVISION EXERCISES 37 1 3. (a) Express each of the following numbers as the product of prime factors : 468, 561, 1547 and 2431. () Find the least square number which is divisible by 7, 8 and 9. 4. Simplify ._||i^ +li of2^. 5. (a) Multiply 1-183 by 2-145 an d divide the result by 0-0845. () How much butter at 2 francs 25 centimes a kilogram would cost 9 francs 90 centimes ? 6. A rectangular plot of building land with a frontage of 34 feet and a depth of 125 feet was let on a building lease at the rate of ^60 per acre. Find the rent of the plot to the nearest penny. 7. Find to three places of decimals the square root of 1314-151. Find a whole number between 1000 and 2000 which is a perfect square and which is divisible by 13. 8. The rateable value of a certain town is ,74360. Find (to a farthing) the smallest rate per pound which must be levied in order to raise a sum of not less than 9. Find as accurately as the data allow, the weight of a cubical block of metal, the edge of which is 3 yards, 2 feet, 3-1 inches long (true to the nearest hundredth of an inch), while a cubic foot of the metal weighs 8-8764 cwt. (true to the fourth decimal figure). In your answer show up all the figures which are accurate but no more. 10. The price of a standard troy ounce of silver on ist January in each of the ten years (1891-1900) was (in pence) 45, 40, 36, 29, 30, 31, 28, 27, 27, 28. Draw a curve showing its value approximately at any time during these ten years. B 1. Find the value of 0-03125 of ^3. 4s. 2. Extract the square root of 7 to 3 decimal places. 3. If the cost of keeping 25 horses for 64 days is ^96, what will be the cost of keeping 75 horses for 28 days at the same rate? 372 A MODERN ARITHMETIC 4. The rainfall on a flat roof 20 ft. long and 12 feet wide is collected in a covered rectangular cistern 5 ft. long, 4 ft. wide and 3 ft. deep ; what depth of water will be found in the cistern after a fall of 2 inches of rain ? 5. If numbers are formed by arranging all the digits 2, 3, 4, 5, 6 in any order, show that each of these numbers when divided by 9 will have a remainder 2. 6. A tradesman bought 5 cwt. of tea for ^37. 6^. &, at what price per Ib. should he retail it in order to make a profit of 25 per cent. ? 7. What rate of interest do I receive if I get %2. IDS. by lending ,500 for 5^ years, simple interest ? 8. Express in decagrams the weight of 447 litres at 19 grains per cubic centimetre, taking i gram=i5 grains, (i litre is I cubi< decimetre.) 9. Find, to the nearest millimetre, the value ol 0-I453X 3-I4I59 of 2 . 5I metres. Io 10. A rectangular piece of paper is measured in inches and i: i centimetres, with the following results : Length, 8-73 inches ; 22-17 centimetres. Breadth, 5-57 inches ; 14-15 centimetres. Find the area of the paper in square inches and in squar centimetres, and hence find the number of square centimetres in i square inch. If the measurements are correct to the nearest figui * in the second place of decimals, to how many places may yoi r results be regarded as accurate ? C 1. The internal measurements of a rectangular cistern are- - length 2-40 metres, breadth 1-22 metres and height i metre. He v many litres of water will it hold ? 2. On what sum will the simple interest for i| years at i\ p :r cent, per annum be ,47. is. 6d. ? REVISION EXERCISES 373 3. Find the cost of papering a room 19 feet long, 16 feet broad and 10 feet high (one-fourth of the wall space being taken up by the fireplace, the doors and the windows) with paper which is 21 inches wide and costs i\d. a yard. 4. The Navy Estimates for the year 1903 were ^35,837,000, and for 1904 they were ^38,328,000. Find, to the nearest tenth, the percentage increase. 5. If 36 horses can plough 54 acres of land in 15 days, how many days will 26 horses take to plough 117 acres at the same rate? 6. A plan is to be enlarged so that a plot of ground which occupied 1-5 sq. in. in the original shall occupy at least i sq. foot in the enlarged plan. If the original measured 6 in. by 4 in., what must the least dimensions of the enlarged plan be? 7. A centimetre being 0-3937 inch, find to four significant figures how many acres are in a square kilometre. 8. The return railway ticket from a suburb to London costs 2s. 2d., and the annual ticket 12. $s. od. How many days in the year must a man travel to save money by taking an annual ticket instead of paying for each day ? 9. (a) One degree of longitude at the equator measuring 365,000 feet, determine the equatorial circumference of the earth in miles. (b] Express as the decimal of 5 furlongs the difference in height between Mount Everest (29,002 feet) and Mont Blanc (15,732 feet). 10. How many parcels each containing i franc and 2 marks can be obtained for ^1063. io.y., when i is worth 25 francs or 2O marks ? D 1. Divide 54076539 by 63, using the method of factors and short division. Give a reason for the method by which you find the remainder. 2. A man has ^108. ly. ^d. Find how many things each costing 5. 4s. 2\d. he can buy, and how much money will remain over. 374 A MODERN ARITHMETIC 3. Find the values of (i) i3 (ii) 230-69x0-0018-^0-059. 4. Find the values of (i) T 5 R of 10 tons 7 cwt, (ii) 3-825 of 3 miles. Express the latter result in miles, furlongs and yards. 5. Find, to the nearest centime, the price of a carpet 6 metres 24 centimetres long, and 4 metres 35 centimetres broad, at 8 francs 5 centimes per square metre. 6. If I pay a cabdriver one penny per minute, and he drives 3^ miles in half-an-hour, at what rate do I pay him per mile ? 7. Find the simple interest on ^3757 for i\ years at 3^ per cent, per annum. 8. Find the value of 35-705682x581-35823-^82-05972 to three places of decimals, using contracted methods. 9. The Navy expenditure is given below for certain years Plot these amounts on a diagram and draw a broken line through the points. Draw also a straight line to lie as near this broken line as possible, and taking this line to indicate what the expenditure would be in 1905-6, and in 1906-7, if there was no change o: plans, estimate the expenditure for each of these years. If the population may be estimated as 44 millions in 1906-7, what wouk the cost per head be in that year ? 1897-8 .... 20,848,863 1898-9 - - - 23,880,875 1899-00 - - - - 25,731,220 1900-1 .... 29,998,529 1901-2 .... 30,981,315 1902-3 .... 31,003,977 1903-4 .... 35,709,477 1904-5 .... 36,889,500 E 1. (a) Express 3 tons 12 cwt. 3 qr. as the decimal of 5 tons. Find the value of 4 ac. 3 ro. 35 sq. po. at ^5. 8^. 6d. pe; acre. REVISION EXERCISES 375 2. Evaluate true to two decimal places : (a) The square root of 87321-7342592. (b) The number of cubic inches in a cubical block of 23-572 inches side. 3. The crop on 17 acres is worth ,140. 5^. at 45^. a quarter. What number of acres will give a crop worth ,154 at 42^. a quarter ? 4. What two figures placed to the right of 785634 will give a number divisible by 97 ? 5. Find the greatest common measure and the least common multiple of the numbers 15496 and 12665, an d show, by general reasoning, that these four numbers form a proportion. 6. The interest of ^25. for 3^ years at simple interest was found to be ,3. 1 8j. qd. ; what was the rate per cent. ? 7. An embankment, 15 feet high and 60 feet long, is built by 30 men of equal strength in 60 days ; another embankment, 20 feet high and 80 feet long, is built by 45 men, also of equal strength, in 64 days. What is the ratio of the strength of the two classes of men ? 8. On i6th December the average price of wheat was 28^. 5^. per 480 lb., of barley 24^. $d. per 400 lb., of oats ijs. nd. per 312 lb. Find, correct to the nearest penny, the price of each per cwt. 9. The following table gives the mileage and annual cost of certain urban main roads in the County of Bedfordshire : Miles. Fur. Ch. Cost. Bedford - - - 9 5 6 ,3000 Biggleswade - - - 10 6 5 ^1575 Dunstable - - -201 ^353 Leigh ton Buzzard - - 8 i 4 ^1225 Luton - - - - 16 2 6 .3300 Calculate the average cost per mile in each case. You may assume that the distances are correct to the nearest chain. A chain = 22 yards. 376 A MODERN ARITHMETIC oles in F 1. Find the value of a farm 125 acres 3 roods 32 sq. pol( extent at ,32. 6s. 8^. per acre. 2. How many bricks 9 inches long, 4 inches wide and 3 inches thick will be required to build a wall 108 ft. long, 8 ft. high and 15 inches wide, if of the volume of the wall is occupied by mortar ? 3. A man bought certain goods of which he sold J at a profit of 14 per cent., $ at a profit of 17^ per cent., and the remainder at a profit of 20 per cent. What was his profit per cent, on the whole ? 4. Given that the mean thickness and diameter of a farthing are respectively and f of those measurements of a penny, find how many farthings will weigh the. same as 48 pennies. 5. Explain the difference between a multiple, a common multiple and the least common multiple. Find the two numbers of which 317 is the G.C.M., 10,461 the L.C.M. 6. If a cubic foot of water weighs 62-426 Ib. and a linear foot is equal to 30-48 centimetres, find the number of pounds in a kilogram, correct to the first decimal place. 7. Explain any method with which you are familiar for shorten- ing the process of obtaining the square root of a number. Illustrate by applying your method to the following example : Find the number of feet in a metre given that i hectare =107642-9934 1 9... sq. ft. 8. Express the difference between 3} and ff 5 as a vulgar fraction. If you were to calculate the area of a circle, using in turn for TT each of these approximations, show that the radius must be as much as 28 cm. for the difference in the results to amount to i sq. cm. 9. If an eight-oared boat is rowed 4^ miles in 21 minutes 36 seconds, find the time, to the nearest tenth of a second, in which it is rowed its own length of 59 feet. REVISION EXERCISES 377 10. One tradesman states his profit as 15 per cent, on the buying price ; another as 15 per cent, on the selling price. Find the difference in their actual profits on goods sold at ,3875. State what you consider the relative advantages and disadvantages of the two methods of computing the profits. G 1. A room is 14 feet 6 inches long, 11 feet wide and 9 feet high. It has a window 8 feet by 4 feet and a door 7 feet by 2 feet 6 inches. Find the area of its walls and the quantity of brown paper required to cover them. The paper is in sheets of 45 inches by 29 inches. 2. A swimming bath, 50 feet long and 25 feet wide, is 7 feet deep at one end and 3 feet 6 inches at the other. Find the number of cubic feet and the number of gallons of water required to fill it to within 6 inches of the top. A gallon weighs 10 pounds and a cubic foot of water may be taken to weigh 1000 ounces. 3. Prove that the least common multiple of two numbers is the product of the numbers divided by their greatest common measure. Find the least common multiple of 1404, 3042, 663. 4. If 20 men, working 6 hours a day, can in u days build a wall mile long ; in how many days of 4^ hours each can a party consisting of 7 men and 10 boys build a similar wall 156 yards long, supposing that 3 men can in a given time do as much work as 5 boys ? 5. If the charge for conveying a party of tourists is two shillings per mile, express this charge in francs per kilometre, supposing that a kilometre is 1093-6 yards and that 25-08 francs are equivalent to^i. 6. In the following advertisement, find the rate per cent, per annum paid by a customer adopting " credit " terms : " Cuckoo- clock offered at 12s. 6d. for cash, or on receipt of $s. and your promise to pay 2 s. 6d. monthly until 1 5 j. is paid." 7. In 1892, Jan. I was a Friday. When did this occur again in that century, and when in the next, 1900 not being a leap year? Give all the steps of your reasoning. 378 A MODERN ARITHMETIC 8. If 675 metres of silk cost 4921 francs 25 centimes, what is the equivalent price per yard in English money? (i metre = 39-37 inches and ,1 = 25 francs.) 9. A table is formed showing the average number of runs per innings of a certain batsman. Compare the effect, in increasing or diminishing the preceding average, of any given score in the 7th innings with that of the same score in the 42nd, every innings having been completed and the average after the 4ist being known to be the same as after the 6th. What is that average, if a score of 88 in the 7th innings adds 6| more to it than the same score in the 42nd ? 10. Find the product of 186-438 and 2-374 true to three places of decimals. If the two numbers given above represent in metres the length and breadth respectively of a rectangular area, true to the nearest figure in the third decimal place, to how many places of decimals can you rely on the product as representing the area, when expressed in square metres ? H 1. A number of books will fill an exact number of shelves with 36, 45, 54 or 60 books on each shelf. What is the least number of books for which this is possible ? If the difference between the greatest and least number of shelves that may be used is 42, what is the number of books ? -1_A+J!__L 2. Simplify 5Q 5Q 2 5Q 3 53 _3___3?.__L 47 5 2 5 3 5 3. What decimal of 365 days, correct to ten places of decimals, is the difference between ^ -of a day and 5 hr. 48 min. 45 sec. ? 4. There were 5852276 names of scholars on the registers in 20513 schools, and the average number of those scholars attending daily was 4890237. Determine (i) what percentage, correct to two places of decimals, the average daily attendance was of the total number on the registers ; and (ii) at the average rate, what should REVISION EXERCISES 379 have been the average daily number of absentees, to the nearest unit, from schools containing 87 and 753 scholars respectively. 5. Find by the contracted method the square root of 3121, correct to within a hundred-thousandth ; and thereby obtain (55) 2 > (5586) 2 , and the first seven figures of the square of 5586591. Show that Jio is less than Z?i and greater than 4 ^43 ; and that 228 1405 it differs from the latter by a quanticy less than 2X228x 1405 6. What rate per cent, per annum does a customer pay, who adopts credit terms in buying the watch referred to in the following advertisement : " 6d. deposit. Watch 6^. 6d. cash. Government stamped case, . . . three-quarter plate jewelled, timed and seven years' guarantee. For advertisement only on receipt of 6d. deposit and payment of 6d. per week, making a total of js. 6d. in all." 7. Assuming that the square of the speed of a falling body is proportional to the height fallen through, and that after falling through a height of i foot the speed is 8-025 feet per second, find, to within one-hundredth of a foot per second, what the speed will be after falling through 873-4 feet. 8. If radium costs 700 per gram, and no milligrams of radium are required to illustrate a lecture, the charge for admission to which is four shillings, how many people would have to be present in order to pay the cost of providing the radium ? 9. The three dimensions of a rectangular block of stone are 12^, 8| and 9^| inches. Give its volume in cubic inches to the nearest integer, first converting the given dimensions to decimals to the necessary number of places. 10. What will it cost to cover the walls of a room 22 ft. 6 in. long, 18 ft. wide and 14 ft. high, with paper 2 ft. 3 in. wide, the price of which is 3.?. 6d. per piece of 12 yards ? I 1. What is the difference in value between a thousand guineas and a million farthings? 380 A MODERN ARITHMETIC 2. Define a measure of a number. Write down all the measures of 363, and the measures which are common to that number and to 231. 3. Express as vulgar fractions in their lowest terms : (i) III ; (ii) Vk-(Sof*T-|l) > ' 4. A man stands by a basket, and ten potatoes are placed in a straight line upon the ground at intervals of 10 yards, the first potato being 10 yards from the man. If he takes them up singly and drops them into the basket, how far must he go ? If there are 100 potatoes similarly placed, but at intervals of i yard, can you find a short way of calculating how far the man will have to go ? 5. Find the squares of 2, 2j, 2 r 4 T and 2}%, as decimals to the nearest thousandth. For each square state its difference from 5, in excess or defect. 6. Find, to the nearest square metre, the area of a garden 26-71 metres long and 18-37 metres broad. If the length and breadth were given to the nearest centimetre, to what degree of accuracy could the area be calculated ? 7. What percentages (to the nearest integer) of the United Kingdom do England, Wales, Scotland and Ireland form? Area of England - - - 32,346,000 acres. Wales - - 4,774,ooo Scotland - - - 19,456,000 Ireland - - - 20,334,000 8. A rate of is. ^\d. in the pound is levied on a district whose rateable value is ,369,782. What sum does this rate produce? 9. ;iooo is advanced to a man who agrees to repay the debt by three equal instalments to be paid at the end of one, two and three years from the date of the advance. The total value of the instalments at the time when the last one is paid, including interest at 4 % per annum allowed on each instalment from the time when REVISION EXERCISES 381 it is paid, is to be equal to the amount of the loan with interest at the same rate for three years. What is the amount of each instalment ? 10. Find the acreage of a field ABCD having four straight boundary lines. E and F are points in AB so that AE = 2-3 chains, EF = 39 chains, FB = 2-2 chains, ED = 2 chains and FC = 2-3 chains. ED and FC are both at right angles to AB. (i chain equals 22 yards. The area of a right-angled triangle is half the product of the sides about the right angle.) J 1. Taking a year to contain 365 days 5 hr. 48 min., calculate the yearly income which averages I farthing per minute. 2. If a tradesman bought 1526 hats at 16^. 6d. each and sold them for cash at a guinea apiece, how many had he left when the cash taken for those sold was equal to the cost price of all the hats? 3. Find to the nearest inch the length of paper which would be used in papering the four walls of a room 21 feet long by 13 feet broad by 8 feet 6 inches high, if no allowance were made for windows, doors, wastage or overlapping ; the breadth of the paper used being 20^ inches. 4. Simplify 2 f&+|i-||of|. 5. Statistics show that of 696419 boys who reach the age of 15 years, 30785 may be expected to reach the age of 85 years. Hence, of 729 boys 15 years old, determine, to the nearest unit, how many may be expected to reach the age of 85 years. 6. Divide 0-1817 by 4-6, and express 17 grams 3 centigrams as the decimal of T.\ kilograms. 7. Find the square root ol T 5 g, correct to the nearest figure in the third decimal place. 8. How much will 465 miles 5 fur. 165 yd. of wire cost at i6s. M. per mile? 382 A MODERN ARITHMETIC 9. What must be the yield per acre of wheat grown on certain land when wheat is worth 2gs. gd. per quarter in order that the value of the crop may be \2\ per cent, greater than when the yield was 31^ bushels to the acre and wheat was worth 34^. per quarter ? 10. Calculate the simple interest on 15637. IQJ. 6d. for the days between 3rd June and 3ist October of the same year of 365 days, at 3^ per cent, per annum, correct to the nearest penny. 11. One lot of articles cost 246. 15^. and another lot of similar articles cost 283. los. If each article cost the same sum, what is the least total number of articles that can have been bought ? K 1. What fraction added to the continued product of f, 2^, 2j (] r will give the continued product of |, 9, , \\\ and f ? 2. Express with a common numerator the fractions 3. The solar year contains 365-242218 mean solar days. Find, to the nearest minute, the error produced in 2000 years by our calendar, which, by means of Leap years, intercalates 97 days in 400 years. 4. A tank one metre deep has a capacity of 35-1 litres. Another tank of the same depth, but having its length and breadth each i centimetre longer, is found to hold 39-2 kilograms of water. Find the length and breadth of the first tank. (A litre is a cubic decimetre ; a litre of water weighs a kilogram.) 5. If it takes 2 days i hour to do 60 miles walking 7 hours each day, how long will it take to do 42 miles walking one hour more each day, but half a mile an hour less ? 6. A square field containing 10 acres is surrounded by a ditch 3 feet deep, 2 feet wide at the bottom and 6 feet wide at the top. How much land is taken up by the ditch, and how many cubic yards of earth had to be dug out to make it? (An acre =4840 square yards.) REVISION EXERCISES 383 7. In a chemical experiment I require to dissolve one gram of copper in acid. I have a reel of copper wire 100 metres long, which weighs 3^ kilograms. What length of wire must I cut off for the purpose ? Mark off the length in question in your answer book, writing your numerical answer below the measured length, calculated to the degree of approximation attainable in measuring. 8. Knowing the, number of pounds in a cubic inch of a sub- stance, you can find the number of kilograms in a cubic centimetre by multiplying by o-4536-f-(2-54) 3 . Express this multiplier as a decimal to three places. If steel weighs 488 Ib. per cubic foot, how many kilograms per cubic centimetre does it weigh ? 9. Find the smallest number that when divided by 21, 24 and 28 leaves as remainder 9, 12 and 16 respectively. 10. A passenger steamer started from Liverpool for New York and maintained a uniform speed for 40 hours, when her engines broke down, causing a delay of 6 hours. After temporary repairs she proceeded on her journey at a rate less than her former rate by 3^ miles per hour, and in consequence arrived at New York 26 hours late. If she had left Liverpool at the reduced rate of speed, and proceeded uniformly without mishap, she would have arrived two hours later than she did. Find the distance between the two ports. L 1. A cubic foot of Canadian elm weighs 0-725 as much as a cubic foot of water ; and a cubic foot of water weighs 1000 oz. : what will be the weight of a beam of Canadian elm 12 feet 6 in. long, i ft. 6 in. deep and i ft. 3 in. thick ? 2. A man buys a parcel of coffee and re-sells it, losing 3 per cent, on the transaction. If he had obtained 14. more, he would have gained 4 per cent. What was the original sum paid for the coffee ? 3. A person gave ^75 for 20 casks of oil, each containing 30 gallons. He sold 5 casks at three shillings per gallon, one cask was stove in and the whole of its contents lost, and 15 gallons were also lost by ordinary leakage. He then sold the remainder 384 A MODERN ARITHMETIC at a price per gallon which made his gain amount to 20 per cent. on the whole transaction. What was his selling price per gallon at the second sale ? 4. What principal will amount to 10,672. qs. 30?. in 7 years and 3 months at 5^ per cent, per annum simple interest ? 5. Define greatest common measure and least common multiple. Resolve 2310, 6552 and 12165 m *-O their prime factors, and thence deduce their L.C.M. 6. For 9 months it costs ,315 to feed 15 horses ; how long can 24 horses be fed for ^448 ? 7. Given that a quadrant of the earth = 10,000,000 metres and that the earth's radius = 3956 miles, find the value of the metre in English measure, correct to the nearest hundredth of an inch, given that the circumference of a circle = 3-i4i6 times the diameter. 8. On a certain map a road 1320 yd. long is represented by a length of i8| inches. Determine the scale of the map. What area on the map would represent an area of ^ sq. mile ? 9. Work out to three significant figures the value of 0-0002938 10. The temperature taken every two hours one day showed : Midnight - - 4i-o 2 p.m. - 5I-2 2 a.m. - 4o-8 4 p.m. 53-o 4 a.m. - 4o-7 6 p.m. - 46- 5 6 a.m. - 39-5 8 p.m. - 46-3 8 a.m. - 4o-8 10 p.m. - 467 10 a.m. - 44- 5 Midnight - - 47-4 Noon - 48-o Draw a curve to show the variation of temperature throughout the day, and estimate the temperature at 3 p.m. Ml. i square metre =10-7643 square feet. Find the length of i decimetre in inches to three decimal places. REVISION EXERCISES 385 2. Working in decimals and by contracted methods, find the price per cvvt. (to the nearest penny) when 23 cwt. 3 qr. 24 Ib. of goods cost ,452. i is. ^d. 3. Find the number of hours and the number of Thursdays from noon on April I2th, 1794, to noon on April I2th, 1894. 4. If a pipe of 9-inch bore discharges a certain quantity of water in 6 hours, how long would 4 pipes of 6 inches bore take to discharge three times the quantity ? (The rates of discharge are as the squares of the diameters of the pipes.) 5. The length, breadth and height of a rectangular room are /, b and h yards respectively. Find formulae for (i) V, the volume of the room ; (ii) A, the total area of walls, ceiling and floor ; (iii) D, the diagonal of the room. Evaluate these when /=8, 6=6, 6 = 4- If 77, a and d denote the volume, area and diagonal of a similar room, each dimension of which is half the corresponding dimension of the larger room, state the relations between v and V, between a and A and between d and D. 6. If 7 rix dollars are worth 2 ducats and 9 ducats worth 4 moidores and 20 moidores worth ^27, how many rix dollars are there in ^72 ? 7. In the following give your reasons, a "yes" or "no" alone being of no value : (a) 15 tons of coal cost 16. ios., and 9 tons cost ,9. iSs. Is the price proportional to the quantity ? (b) The base of one equilateral triangle is 2 in. and its area 0-866 sq. in., the base of another is 4 in. and its area 3-464 sq. in. Are the areas proportional to the bases ? (c) ^50 put out at interest amounts in two years to ^54, in three years to ^56 and in five years to ^60. Is the amount proportional to the time ? Is the interest propor- tional to the time ? 8. A swimming bath holds 37,500 gallons. Find the cost of filling it at yd. per thousand gallons. If, however, the charge is based on the reading of a water meter J.M.A. 2B 3 86 A MODERN ARITHMETIC which may be wrong by 10 per cent, of the true value, how much too great or too little may the charge be ? 9. Find to within i mm. the length and the breadth of a rectangular table whose area is 5-5 square metres and which is twice as long as it is wide. N 1. Prove that the least common multiple of two numbers is obtained by dividing the product of the numbers by their greatest common measure. The least common multiple of two numbers is 16830 and their greatest common measure is 198 ; find the numbers. Show that there are two answers. 2. The average number of thousands of depositors in the Post Office Savings Bank for the years 1891, 1892, 1893, 1894 was 5606 (thousands). The number of depositors in the year 1896 was 6-6 per cent, higher than in 1895, while the average for the six years 1891 to 1896 was 5945. Find the numbers of thousands of depositors in each of the years 1895 an d 1896. 3. The net prices of goods described in a catalogue are 1 5 per cent, less than the catalogue prices, while a further discount of 5 per cent, on the net prices is given when payment is made within a month. A customer, however, obtains 20 per cent, off the catalogue price, and finds that by so doing he has gained 4. los. 6d. What was the catalogue value of the goods he bought ? 4. The outer dimensions of the concrete foundations of the walls of a building are : length, 10 metres 25 centimetres ; breadth, 9 metres 38 centimetres ; and depth, 85 centimetres. The inner dimensions are : length, 9 metres 45 centimetres ; breadth, 8 metres 58 centimetres ; and depth, 85 centimetres. Express in steres the amount of concrete required. 5. The dimensions of a floor are 42 ft. 9 in. and 35 ft. 9 in. What would be the error in the calculated area if the dimensions were taken (i) as 42 ft. and 35 ft. respectively, (ii) as 43 ft. and 36 ft. respectively ? REVISION EXERCISES 387 6. Express the difference between the Julian year (365^ days) and the Gregorian year (365 days 5 hours 48 minutes 47-6352 seconds) as the decimal of i day. 7. A ladder is 2 ft. 6 in. wide at the base and i ft. 4 in. at the top. If there be 23 rungs, and if the middle rung be equi-distant from the base and top of the ladder, find the total length of wood which has to be used for the rungs. 8. I wish to fix the size of a circular enclosure which I may not enter ; I therefore note the following points : It is surrounded by a footpath 10 feet wide, and four lamp posts stand along the outer boundary of the footpath at the corners of a square. Each side of this square cuts from the enclosure a strip whose greatest width is equal to the width of the path. Find the diameter of the enclosure by calculation and also by drawing. For your drawing take a circle of radius 4 inches for the outer boundary of the footpath. 9. Find the rate per cent, per annum, simple interest, that a borrower pays, according to the following scheme : " Loan Fund Association. Money promptly advanced on personal and other security. ,20, 12 monthly repayments of i. 17^. %d" 10. A rectangular casket (with a lid) 1 1 inches long, 7 inches wide and 6 inches deep, outside measurements, is made of metal of a uniform thickness of fths of an inch, and plated inside and out with silver ylfyth of an inch thick. Find the weight of silver on the casket. (A cubic inch of silver weighs 4-5456 ounces troy.) 59. LOGARITHMS. Introductory. A logarithm is defined formally as follows : The logarithm of a number to a given base is the power to which the base must be raised to equal the number. Thus, since 2 3 = 8, 3 is the logarithm of 8 to the base 2, or, more briefly, 3 = Iog 2 8. 3., <.U 388 A MODERN ARITHMETIC Similarly, 5 = log 3 243, for 243 = 3 5 . The application of logarithms will be seen later. By the use of logarithms, processes of multiplication become processes of addition, processes of division become processes of subtrac- tion and processes of evolution and root extraction become processes of multiplication and division respectively. Clearly 3 3 x 3* = (3 x 3 x 3) x (3 x 3 x 3 x 3) = 3?. Also, x* x x 4 = three x's all multiplied together, and the answer multiplied by the product of four more x's = x 7 , where x stands for any whole number. The result can be extended, and where the powers or indices m, and n, are any whole numbers. Frequently fractional indices have to be employed ; thus : x% has the same meaning as the square root of x, y* M cube y, z^ ,, fifth ,, z, and so on. a? can be regarded either as the square root of the cube of a, or the cube of the square root of a. Some of the following exercises are designed to show that the relation x x x _ x m+n is true when m and n are fractional. Also that x m + x n = x m ~ n . EXAMPLE i. 256* X 256* = 22 X 2^= 2 4 X 2 2 = 2 6 = 256^. EXAMPLE 2. 15625^ x 1 5625^= 5* x 5 = 5 2 x 5 3 = 2 5 = 15625^. EXERCISES 389 EXERCISES LXXXV. a. 1. Find the products of the following, expressing the results as powers of the numbers given : () 3 2 x 3 3 J (*) 4 2 x 4 4 J (c) io 3 x lo* ; (d) 4 3 x 4 10 ; (e) ioo 3 xioo 2 ; (/) 2 5 ^X25 32 ; (g) (0 <f x ^ ; ( I) X U Fill in the missing parts in the following equations, working on the lines of examples i and 2 on p. 388 : 2. 1024^x1024* =io24<-"->. 3. 1024^x1024^ =io24 ( ---->. 4. T024 Tor X 8. (icxx>oc)coo)^-h(ioooooooo)' s =(ioooooooo)'----'. Since therefore 2X2 = " x 2" = Obtain relations similar to the above in the following cases 9. 10. 11. 12. (3 3 6 x3 n io 7 xio 9 =3 16 . =io 16 = (o . 6 13. 14. 15. 16. 17. 18. 8 13 -8 5 I3 30_, 3 ir I0 12 -I0 9 = io 13 . = io 3 . Extract the 12 th root. u th 30 th 17 th 40 th 5o th u th n a xn b 390 A MODERN ARITHMETIC Since n a + n b = n a ~ l> t it is clear that n a ~ 6 = - 5 . n EXERCISES LXXXV. b. Oral. Change into or from a fractional form : 1. io 13 - 3 . 2. 10*-*. 3. ill. 4. 6. ^. 7. (ioo) 12 - 0125 . 8. N 10 - d p 10. . 11. C 12. . 13. T. 14. 2i. 15. *". (o-oi) 4 It has probably been noticed that ff -i- * = n x ~ x = n by following out the rule, whence =i. This is of great importance. i Hence also -s = -5 = ^~ a = ^~ a - n* n a EXERCISES LXXXV. c. Oral or otherwise. Express 1. ^ as a power of 2. 2. (a) o-oo i, () ioo, (^r) o-oi, (d) Toksv as powers of io. 3. (a) ^5, (b) 125, (r) 0-008, (d) 0-04, (^) T 5^s as powers of 5. 4. (a) 0-25, (d) 0-125, fc) ToVf' (^) 2r,ff as powers of 2. 5. (a) & (b) fr, (c) T^y as powers of 3. EXERCISES 391 Between what two consecutive powers (positive or negative) of 10 do the following lie : 6, 782. 7. 0-063. 8. 0-0007. 9. 973-2- 10. 108932. 11. o-ooooo6. 12. 699999. 13. 82-0063. 14. 5 8'- 15. ft 16. if* 17. ^f*. 18. 1938"- 19. 1-0683. 20. 9^. 21. 22. 0-000007. 23. 32 I0 8 * EXERCISES LXXXV. d. Find the logarithms or 1. (a) 4, (*) I (4 0-0625, (</) 256, (*) 64, (/) g V, Or) 1024, (A) 0-25 to the base 2 ; z'.. find the power to which 2 must be raised to yield the given number. 2. (a) 3, (*) 3 3 , (0 , (*0 243, (') 8\, (/) ^r, te) 27 to the base 3. 3. () 2, <*) 8, (^) S, (O ^ to the base 64. 4. (a) 5, (^) 0-2, (c) 0-04, (rt 7 ) 125, 0) 625, (/) ^Vs to the base 5. 5. (a} }, (b) 49, (c) 343, ('0 JT!^ W i^ to the base 7. 6. (a) 5, (6) 125, (tr) F | , (d) M ^ to the base 25. 7. (a) 3, (*) 27, (c) 9, (rf) t, (') ffh to the base 8l - 8. (a} o-i, (^) o-o i, (c} TOGO, (</) 100, (e) 10, (/) 10000, (g) o-ooooi to the base 10. 9. Express the following relations otherwise than in logarithmic form, proving the results where not obvious : (a) Iog 10 o-i =-i, (b) Iog 10 ioo =2, (c) Iog 10 100000= 5, (d) Iog 3 8i =4, (e) Iog 4 o-i25=- 1-5, (/) Iog 2 1024 =10, (^) log 10 o-ooi = -3, (//) log 5 o-oo8=-3, (i) log*.* 5 =2, (7)log 4 8 =1-5, 0&)log 16 8 =|, (/)log 8 i6 =|. 392 A MODERN ARITHMETIC Proofs of Fundamental Processes. Proofs will now be given of the fundamental processes i the application of logarithms. (a) \og b m + log b n = \og b (m x n\ (b) logj m - Iog 6 n = log b (m + n\ If x = \og b m, then m = b x , and hence mxn = b x xl^ = b x+y ; '. \og b mn = x+y =logw + log^. Similarly, m^n = b*-V = b* Again, *~ y As a particular case of multiplication or of root extraction it will be clear that In practical applications the base is always 10; it will there- fore be assumed in future that the base is TO, if no specific mention is made of the base. Thus, by log 5 will be meant the logarithm of 5 to the base 10. Tables of logarithms are published, and a table is given at the end of the book, but it is interesting to obtain the logarithms in certain cases, even if the methods employed are not general methods and are only applicable in special examples. LOGARITHMS 393 The logarithm of any power of TO can be obtained by inspection. Thus: Iog 10 i=o for io=i, Iog 10 io = i 10! = 10, 10g 10 TOO = 2 10 2 =IOO, Iog 10 iooo = 3 i o 3 = 1000, and so on. Also log 0-1= -i io~ 1 = = 0-1, i i lOg 0-01= -2 10 * = ; = = 0-01. I0 2 100 The logarithm of any number, pure or mixed, between i and 10 lies between o and i ; that of any number between 10 and 100 lies between i and 2, and so on. EXERCISES LXXXV. e. Oral. State the integers (positive or negative) between which lie the logarithms of the following numbers : 1. (a) 15, ()225, (c) 7, (d) 18625, (^989256, (/)7-9234 5 Of) 79-234- 2. (a) 0-002, (b) 0-003, (c) 0-04, (d} 0-00006. 3. (a) 0-5632, (b) 0-05632, (c) 1983, (d) 198. 4. (a) 3892, (b} 0-3892. Methods of obtaining Approximations to log 2, log 3, log 7. EXAMPLE i. To find an approximation to log 2. 2 lies between 10 and lo 1 ; log 2 lies between o and i. 2 2 lies between 10 and lo 1 ; .'. 2 log 2 lies between o and i, and log 2 lies between o and 0-5. 394 A MODERN ARITHMETIC 2 3 lies between 10 and lo 1 ; .'. 3 log 2 lies between o and i, and log 2 lies between o and 0-333... Again, since 2 10 =io24, and is greater than 1000, .'. log 2 10 = log 1024, and is greater than log io 3 , i.e. >3; whence log 2 > 0-300 and < 0-333. EXERCISES LXXXV. f. 1. By considering the logarithms of the numbers in the first column, supply answers to the queries in the accompanying table as in Example (i) above : fa) W Or) 2 3 log 2 is greater than and less than 0-333 2 io 55 11 11 0-300 11 11 0-4 2 13 >l 11 11 ? 11 11 ? 2 33 51 55 55 \ 11 11 \ 3 2 log 3 55 5) p 11 11 p 3 15 55 55 ? 11 11 p 3 23 55 55 5) p 11 11 P 3 W 55 55 55 p 11 11 p 7 6 log 7 55 55 p 11 11 p 7 u 55 5> 55 p 11 11 ? Write down as accurately as you can the value of log 2, log 3, log 7, obtained from the above table. (More accurately log 2 =0-30 103... , log 3=0-47712..., log 7 =0-845 1....) 2. Hence obtain (a) log 4, (d) log 5, (c) log 6, (d) log 8, (e) log 9. LOGARITHMS 395 EXERCISES LXXXV. g. Graphic. 1. Draw a graph in which horizontal distances represent the numbers i to 10, and vertical distances the corresponding log- arithms. (Take the lengths representing units to be in the ratio of i : 10; for example, log 2=0-30103... ; the vertical distance would be 10x0-30103 and the horizontal distance 2.) 2. Find by your graph the logarithms of 3-6, 4-2, 1-9, 7-3. Also the numbers the logarithms of which are 0-32, 0-45, 0-78. 3. The table below gives the logarithms of the numbers quoted. Draw suitable graphs showing how the logarithm changes with the change in the number. No. 600 601 602 603 604 log 2-7782 2-7789 2-7796 2-7803 2-7810 No. 605 606 607 608 609 log 2-7818 2-7825 2-7832 2-7839 2-7846 No. 8-00 8-01 8-02 8-03 8-04 log 0-9031 0-9036 0-9042 0-9047 0-9053 No. 8-05 8-06 8-07 8-08 8-09 log 0-9058 0-9063 0-9069 0-9074 0-9079 Make what deduction you can from your result. 4. The table be4ow gives the difference in the logarithms for the same difference in the number. Plot graphs showing how the 39 6 A MODERN ARITHMETIC differences in the logarithms vary (a) with the number, (b) wit the reciprocal of the number : Numbers. Approximate difference in logarithms. 100 and 101 0-0043 200 201 O-OO2I 3 3 01 0-0015 400 401 O-OOIO 500 501 0-0008 600 60 1 0-0007 700 701 0-0006 800 80 1 0-0005 900 901 O-OOO5 60. LOGARITHMIC TABLES AND INSTRUMENTS. To find the Logarithm of a Number. EXAMPLE i. To find log 575-3. Since 575'3 = i 2 x 5'753> log 575-3 = log io 2 + log 5-753, but Iogio 2 =2 and 5-753 >~i<io; .". 5*753> 10 < ro 1 . Hence log 5-753 lies between o and i, and .'. log 5-753 is a pure decimal. The whole number, that is 2, is called the characteristic, the pure decimal ,, mantissa. The mantissa alone is given in the tables, and without the decimal point ; the characteristic is seen on inspection. In the table of logarithms, begin by looking down the first column for 57. The corresponding line is as follows : No. 57 o i 7566 2 3 4 5 7597 6 7 7612 8 9 7627 Difference columns. i 2 2 3 2 4 3 5 4 6 5 7)8, 7559 7574 7582 7589 7604 7619 5^7 LOGARITHMIC TABLES AND INSTRUMENTS 397 we add on the corresponding figure in the difference column : here i ; The meaning of this line is that o = o-7559, ^5-71=0-7566, Iog572 = 0-7574 and so on. To obtain log 5-70 5'? 1 ' etc., whence log 5-701 = 0-7559 + o-oooi = 0-7560, so log 5-707 = 0-7559 + 0-0005 = -75 6 4> and in particular Jog 5-753 = -7597 + 0-0002 = 0-7599 ; hence log 575-3 = 2-7599. EXAMPLE 2. To find log 0-005754. 0-005754 = 5-754 -MOOO -3 = 5-754* io ~ 3 = log 5-754- 3; but log 5-754 = log 5-75 + difference for 4 = 0-7597 + 0-0003 = 0*7600 and log 0-0057 54 = 0-7600 - 3. This is written, however, in the form 3-7600. To find the Number corresponding to a given Logarithm. EXAMPLE. Find the number, the logarithm of which is 3-8828. By inspection it is seen that the logarithm is that of a number between io 3 and io 4 and = io 3 times the number the logarithm of which is 0-8828. Looking through the table of logarithms, the following line is found : 8820 3 8825 6 8842 8854 9 8859 Difference columns. 0-8828 lies between 0-8825 and 0-8831, and therefore between log 7-63 and log 7-64. But 0-8828 = 0-8825 + 0-0003. Look at the difference column ; it is seen that 0-0003 ' 1S tne difference corresponding to a 5 or 6 ; .'. 0-8828 = log 7-636 or log 7-636. 398 A MODERN ARITHMETIC The ambiguity, that is, whether the third decimal figure is 5 or 6, is due to the fact that the logarithms are approxi- mate only. Multiplication by Logarithms. EXAMPLE. Evaluate (0-19 x 132) by logarithms. log (0-19 x 132) -log 0-19 + log 132 =--1-2788 + 2- 1 206 = 1-3994 = log 2 5 -08,; .'. 0-19 x 132 .= 25-08. Division by Logarithms. EXAMPLE. Find the density of a substance, 973-2 c.c. of which have a mass of 9983 grams. (Density = mass -f- volume. ) Density (D) = 9983 -=- 973-2. log D = log 9983 - log 973-2 = (3-9991 4-0-0001) - (2-9881 +0-0001) 1*01x0 From tables, = I -0086 + 0-0024 = log 10-26 ; .*. density = 10-26 gr. c.c. units. From tables, log 9-98 = 0-999 1, diff e for 0003 = 0-0001 ; log 9732 = 0-9881, diff e for 2 = 0-0001. log 1-02 = 0-0086. 24 = difference for 6 nearly. Involution by Logarithms. EXAMPLE i. Evaluate (13-27)* by logarithms. 13-27 = 3(1-1206 + 0-0023) = 3-3687 = 3-3 6 74 + 0-0013 = log 2337; LOGARITHMIC TABLES AND INSTRUMENTS 399 EXAMPLE 2. Evaluate (0-0864)* by logarithms. log (0-0864)* = 4 log (0.0864) = 4(2-9365) (NOTE. ^-9365 means = 5-7460 +0-9365-2, = 5-7459 + 0-0001 and 4 times this = log 0-00005 57 1 ; =3-7460-8 /. 0-0864 = 0-00005571. =0-7460 + 3-8 = 0-7460-5.) Evolution by Logarithms. EXAMPLE i. Evaluate ^897600. If X stands for the required answer. log X = J log 897600 = 1-9844 = 1-9841 +0-0003 = log 96-45 ; .'. ^897600 = 96-45. EXAMPLE 2. Evaluate \/o-oooo8976. If X stands for the required answer, log X = \ log 0-00008976 = i.(6 + 1 .953 1 ) Note carefully the alteration = 2-6510; f 5 ,t 6 + 1 * so that the ' X = 0-044.7 7 negative characteristic can be divided exactly by 3. EXERCISES LXXXVI. a. NOTE. Wherever possible make at the beginning a mental approximation to the first, or the first two, significant figures. 1. Calculate by logarithms the weights of the following : (a) Leaden block, length 98-34 cm., width 34-89 cm., thickness 16-03 cm - (density of lead 11-36). (b) Copper plate, length 4-312 metres, width 2-813 metres, thick- ness 0-0134 metre (density of copper, 8-781). 400 A MODERN ARITHMETIC (c) Copper cube, edge 35-35 cm. (d) Steel bar, 8-23 ft. long, 0-0864 ft- square in section (weight of i cub. ft. of steel 499- 1 lb.). (e) Copper wire, 3061 metres long, radius of section 0-000234 cm., 77 = 3-142 (density of copper 8781). (/) The air in a room 19 ft. 4 in. long, 18 ft. 6 in. broad, 1 1 ft. 9 in. high (density of air 0-001291 x 62-32 in ft. lb. units). () 1894 bricks, the weight of a single brick being 8-23 lb. (k) A seam of coal, 3984 yd. long, 1764 yd. wide, average thick- ness 1-032 yd., assuming i cub. ft. of coal to weigh 78-26 lb. 2. If a metre be 39-37 in., calculate by logarithms : (a) The number of metres in a mile.- (b) yards in a kilometre. (c) cm. in 23-81 ft. (d) cm. in 1064 inches. (e) inches in 18-62 metres. (/) 5 890 x i o~ 10 metres. (g) sq. inches in 18 sq. metres. (h) ,j cub. inches in i cub. decimetre. 3. Using the table below, express (a), (c), (h) and (/) as percentages of the total amount of material in solution : The Material in Solution in i cub. mile of average River Water. Tons in Cubic Mile. (a) Calcium Carbonate - - 3267 x io 2 . (b) Magnesium Carbonate - II29XIO 2 . (c) Calcium Sulphate - - - 3436 x io. {d} Sodium Sulphate - - 3180x10. (e) Potassium Sulphate - - 2036x10. (/) Sodium Nitrate - - - 2680 x io. (g) Sodium Chloride - 1666 x io. (h) Silica - - - - 7458 x io. (z) Ferric Oxide - - - - 1301 x io. (j) Alumina- - - - 1431x10. (K) Organic Matter - - 7902 x io. LOGARITHMIC TABLES AND INSTRUMENTS 401 4. Using the table below, find the volume of a ton of water at a temperature of (a) 32 F., (I)) 212 F. : Weight of Water at Different Temperatures. Temperature. Weight of i cub. ft. -32 Fah. - - 62-417 lb. 40 - 62-423 50 - - 62-409 60 - - 62-367 Temperature. 70 Fah. - 80 - 90 - 212 Weight of i cub. ft. - 62-302 lb. - 62-218 - 62-II9 - 597 5. Employ logarithms to complete the following table : Quantity of Rainfall corresponding to given depth. Tons per sq. mile. Imperial gallons per acre. Depth of rainfall. (a) 3626 ? ? (*) 10877 ? ? (*) 14503 p ? 6. If the matter transported by the rivers mentioned below has an average density 2-23 times that of water, find in each case the number of cubic miles of rock transported per annum. Dissolved Mineral Matter transported by certain Rivers. (a) Mississippi, (b) Danube, (c) Nile, (d) Rhone, - (e) Rhine, (/) Thames, 1 128 x io 5 tons per annum. 2252 xio 4 1695 x io 4 8290 x io 3 ., 5817 xio 3 548 x io 3 7. Find the weight of 172-5 ft. of cast iron piping, outside diameter 14-00 in., thickness of metal 0-625 in., the weight of a cubic foot of cast iron being 474-4 lb. (^ = 3-1416). J.M.A \ 2C 402 A MODERN ARITHMETIC 8. Supply the missing portions of the following table : 00 Diameter of c'rcle. Area of circle. Circumference of circle. O-O24 p 0-006940 O-OO6O82 ' 9. Find the lengths in cm. of the sides of the following cubes : (a) Mass, 8967 grams, density 2-312. (*) 0-006356 8-834. (c) ii84Xio 3 7-726. 10. Evaluate the following : 00 (33-6S5) 4 , (J) (38-125)*, (c) (8763 x io- 3 )* (rf) (0-0008634)?. Evaluation of Formulae. The following solutions are examples on the evaluation of formulae, empirical (that is, based on experience or observa- tion) and otherwise. EXAMPLE r. In blasting with gunpowder, the number of ounces of powder required to blast any rock is given by P, where p = XlJ L being the length of the line of least resistance, X a constant. If X be 4-062, calculate P when L = 4-336 //. Since P = --, log P = log X + 3 log L - log 8 (Note a proceeding = log 4-062 + 3 log 4-336 -Jog 8 frequently employed, = 0-60874 + 3 (0-63709)^+ 1-0969 1 i.e. of subtracting log 8 = 0-60874+ 1-91127 +1-09691 from o at once, so that = 1-61692. the further process is P = 4i-4o addition.) 41-40 oz. of powder will be required. LOGARITHMIC TABLES AND INSTRUMENTS 403 EXAMPLE 2. The load, a solid column or pillar of cast iron can support^ is frequently calculated by the formula W being the weight of the load in tons, D the diameter in inches, L the length of the column in feet. Calculate the load which can be supported by a pillar 7-3 ft. high and 12 in. in diameter. By logarithms. log W = log 14-9 + 3-76 log D- 1-7 log L 1-07918 0-86332 = log 14-9 + 3-76 log 12- 1-7 log 7-3 = 1.17319 + 3-76(1.07918) -i-7 (0-86332) _ = 1-17319 + 4-05772 + 2.53236 = 3.76327. 3-23754 0-86332 75543 0-60432 6475 4-05772 -46764 A load of 5797 tons could be supported. EXERCISES LXXXVI. b. 1. In blowing up walls, soldiers sometimes calculate the necessary charge of powder by the formula P = tfxl_ 3 , where P is the charge in pounds, L the half thickness of the wall in inches, a a constant depending on the nature and so on, of the wall. Calculate P when ^ = 0-1672, L=8324. 2. The deflexion in inches produced by a load uniformly distributed over a beam or girder of rectangular section is given by I2WL 3 where W = the load in tons, L = length of beam in feet, B the breadth in inches, T the thickness of the beam in inches, while E depends on the nature of the substance of the beam. In an iron beam in which E = i-32X io 4 , the length was 18 ft., breadth 12 in., thickness 9-86 in., find the deflexion due to a load of 126 tons. 4 o 4 A MODERN ARITHMETIC 3. The tractive force of a locomotive is given by D 2 PL = W ' where T is the tractive force in pounds weight, W the diameter o the driving wheel in inches, P the mean pressure of the steam in the cylinders in pounds per sq. in., L the length of the stroke in inches, and D the diameter of the piston in inches. The diameter of a driving wheel of a locomotive is 5 ft., the mean pressure is 65 pounds per sq. in., the length of stroke 27-3 in. and the diameter of the piston 10-6 in. Find the tractive force. 4. The number of cubic feet of water discharged over a sill is given by C = 2i 4 V7T a , if the water be otherwise at rest, C being the number of cub. ft. discharged over each foot width per minute, H the height of the water surface above the sill in feet. Calculate the discharge from each foot of width when the sill has a depth of (a) 10-3 in., (&) 6-4 ft., (c) 9-5 ft. 5. The number of gallons of water delivered per hour in pipes is sometimes calculated by N/(l5DyH ~ L ' where G = no. of gallons delivered per hour, H =head of water in feet, D = diameter of pipe in inches, L = length of pipe in yards. (a) Establish a similar formula where all the lengths are expressed in centimetres, but where G is expressed in litres. (Given I lit. =0-2201 gal. and i metre = 39-37 in.) (b) Find the diameter of the pipe if the number of gallons discharged per hour is 2161, the head of water 150 ft. and the length of pipe 1200 yd. 6. A formula sometimes used in calculating the diameter of a single acting pump used in raising water is LOGARITHMIC TABLES AND INSTRUMENTS 405 where L is the length of the stroke in feet, G the number ot gallons to be delivered per minute, N the number of strokes per minute, D the diameter of the pump in inches. Calculate (a) the number of gallons raised per minute when the length of stroke is 3-24 feet, the diameter of the pump 1176 in. and the number of strokes per minute 1 1 ; (b) the diameter of the pump capable of raising 120 gallons per minute when the length of the stroke is 2-94 feet and the pump is working at the rate of 13 strokes per minute. 7. The pressure on a dock gate is given by P= Ax 0-2780, where P is the force on the gate in cwt., A the area exposed to water in square feet, D the depth of water on the gate in feet. Find (a) the total force when the area exposed is 565 square feet and the depth 24 feet 9 inches ; (b) the depth if the force is 31-64 cwt. and the area 616 square feet. 8. The necessary diameter of a balloon, in order that it may be able to raise a given weight, can be calculated by -V- Vo-C2' w 236 (A -G)' where D = diameter of balloon in feet, A = weight in Ib. of a cubic foot of air, G = weight in Ib. of a cubic foot of the gas used in filling the balloon, W = weight raised in Ib. (including the weight of the balloon itself). If A=o-o8o72, 0=0-0056, calculate (a) the diameter in order that the balloon may be capable of raising 12-5 cwt.; (b) the weight that could be lifted by a balloon of 35-5 ft. diameter. 406 A MODERN ARITHMETIC 9. In warming a building by hot water pipes, the requii length of 4 in. pipes is based on the formula _ x 0-00450, P-T where L=the length of the pipes in feet, P = the temperature (F.) of the pipes, T= required in the building, /= ,, external air, C = number of cub. ft. to be warmed per minute. Find the number of feet of piping required when P=i2O F., /=4o-5 F., T = 6i-5 F. and = 35-6 x io 2 . 10. Calculate the force exerted per sq. ft. by wind blowing directly on a surface according to the formula P = 0-002288V 2 , where P is the pressure in Ib. weight per sq. ft.. V velocity in feet per second. Velocity in ft. per sec. Description of Wind. (a) 29-3 Brisk gale (*) 73-3 Storm 0) 146-6 Hurricane 11. The elastic force of steam expressed in inches of mercury is given by /T+ioo p_ where T is the temperature (F.) of the steam, C is 177 and F is the elastic force. (a) What is the temperature if the elastic force is 60-2 in. ? (b) What is the elastic force when the temperature is 279 F. ? 12. The number V of cubic feet of steam obtained from i cub. ft. of water is given by 20559 ~ 0-941 ' LOGARITHMIC TABLES AND INSTRUMENTS 407 where P is the pressure in pounds weight per sq. in. (including the pressure of the atmosphere). Calculate (a) V if P be 14-706, (6) P if V be 103-6. 13. The number M of miles per hour travelled by a locomotive P.D "56-0238' where P = speed of the piston in ft. per minute, 8= stroke of the engine in ft., D = diameter of the driving-wheel in ft. Complete the following table : Diameter of Driving-wheel. Piston Speed. Speed of Engine. Stroke in inches. (a) 3 1867 60 p (*) p 1232 60 22 to 4-5 809 p 26 NOTE. Other exercises on the use of logarithms are given in connexion with Compound Interest and Areas. The Slide Rule. EXERCISES LXXXVI. c. Practical. If a line AB be drawn to represent the logarithm of any number m, and another BC to represent the logarithm of any other number , then a line equal in length to (AB+BC) will represent the logarithm of the product m x n of the two numbers. 1. The common logarithms of the numbers 2 to 10 are as follows : Number - 2 3 4 5 6 7 8 9 10 Logarithm 0-3OI 0-477 0-602 0-699 0-778 0-845 0-903 Q-954 i 4 o8 A MODERN ARITHMETIC On a scale in which 400 mm. represents the logarithm of 10, i.e. i, the preceding logarithms would be represented by the following lengths : Length in mm., 120, 191, 241, 280, 311, 338, 361, 382, 400. Calculate to the nearest millimetre the lengths which represent the logarithms of i-i, 1-2..., 2-1, 2-2..., 9-1..., 9-9 (use a table of logs.). 2. On squared millimetre paper construct a scale, in which are represented, as in Ex. 1, the logarithms of the numbers I, i-i, ... , 9-9, 10. Make the divisions of the scale 3 cm. in length. Care- fully cut the scale into three others, each of the same length but i cm. wide. The scales can now be used as rough slide rules, and the following exercises may be worked either with these paper scales or a more carefully prepared slide rule. 3. Place two of the scales alongside one another ; slide one scale B along the other A until the division marked i in B is opposite to the division marked 2 in A. See if the divisions on A are marked with numbers twice as great as those which mark the opposite divisions in B. EXAMPLE i. Evaluate 1-4 x 8-5 by means of the slide rule. FIG. 72. Move B along A until the division corresponding to 1-4 in A is opposite to the division marked i in B (Fig. 72). Note the LOGARITHMIC TABLES AND INSTRUMENTS 409 reading on A opposite to 8-5 on B ; it is seen to be between 1 1 -8 and 12, and estimated at 11-9 (Fig. 72). EXAMPLE 2. Evaluate 46 -=- 3-4. It is important to note that in many cases no notice of the decimal point is taken in the work with the slide rule itself; its position is determined by inspection. A 3 14 15 D Place division 3-4 in B opposite to division 46 in A, read off the division on A opposite division i in B; this reading gives the result of the division, and is seen (Fig. 73) to be 13-5. EXAMPLE 3. Evaluate 58 x 1-3 -7-7 -9. FIG. 74. (a) Set division i on B opposite division 1-3 on A, move the hair slide until the hair is over the 58 mark on B (Fig. 74). 4io A MODERN ARITHMETIC e hair (b) Move the scale B until the 7-9 mark is under the (Fig. 75). Note the reading on A opposite the mark i on B ; this is seen to be about 9-5. FIG. 75. EXAMPLE 4. Evaluate Method (a). Using only scales A and B. Place the hair slide over division 78 on A, slide B until the division on B underneath the hair is the same as the division in A opposite to the mark i on B. The reading 8-83 gives the square root (Fig. 76). FIG. 76. Method (b\ Using a separate scale. Construct a new scale (this will probably be already part of the slide rule) in which all the scale divisions are exactly double the size of the corresponding divisions on A or B; place this scale D against either A or B ; if the first marks (i) LOGARITHMIC TABLES AND INSTRUMENTS 411 are opposite each other, the numbers on D are the square roots of the corresponding numbers on A (or B). In an actual slide rule, these scales would be as on C and D in Fig. 76. EXAMPLE 5. Evaluate % FIG. 77. Set the hair slide over 76 in A; slide B until the reading on D opposite the i on B or C is the same as the reading on B under the hair (Fig. 77) ; this reading 4-24 gives the cube root. EXERCISES LXXXVI. d. Evaluate the following, as accurately as the slide rule permits, and verify by ordinary calculation : 1. 1-3x1-4. 2. 2-5x1-3. 3. 4-7x1-22. 4. 5-4x1-8. 6. 1 13^- 3-4- 7. 29^-7-4. 8. 3-5-^2-2. 10. (8-9-^ 15)-=- 3-2. 11. 76^(3-1x4-1). 13. \/7 x 4-2. 14. \/9-3^-2-4. 16. X/8-9X3-2. 17. V7. 19. VT^M. 20. \/8-2X2-i. 18. N.B. The slide rule should, however, be frequently employed in approximations in general work ; accordingly, only a few typical exercises are given here. 412 A MODERN ARITHMETIC EXERCISES LXXXVI. e. Graphic. 1. The following table gives the volumes and diameters of number of spheres. Draw a graph, plotting the logarithm of th( volume against the logarithm of the diameter ; measure th( increase in the logarithm of the volume for each unit increase ii the logarithm of the diameter ; show from your result that th< volume is proportional to the cube of the diameter : Volume 0-524 4-19 I4-I 33-5 65-5 113-1 179-6 268 382 cub. in. Diameter I 2 3 4 5 6 7 9 in. 2. The accompanying table gives the velocity acquired by bodies in falling from various heights. Plot the logarithms of the velocity against the logarithms of the fall. Deduce from your graph that the velocity is proportional to the square root of the fall : Fall in feet - i 10 20 30 40 50 60 70 Velocity in feet per sec. - 8 25 36 44 51 57 62 67 3. The table below gives the relation between the maximum supply of gas through a certain length of piping and the diameter of the piping, the pressure being the same in each case. Prove that the supply squared is proportional to the diameter raised to the 5th power : Diameter of pipe in inches - I 2 3 4 5 6 Supply in cub. ft. per hour - 67 377 1039 2133 3730 5880 4. The following table gives the mean distances from the sun of the chief large planets ; also their periods of revolution around the sun in days. Draw a graph connecting the logarithms of the mean distance trom the sun with the logarithm of the period. Make what inference you can from your result, and find LOGARITHMIC TABLES AND INSTRUMENTS 413 (a) the period of Neptune distance 2791 x io 6 miles ; ( distance of an asteroid, the period of which is 1200 days: the Planet. Mean distance in millions of miles. Period in mean solar days. Mercury 36-0 88 Venus 67-2 225 Earth 92-9 365 Mars - HI'S 686 Jupiter 483-3 4332 Saturn 886-0 ic-759 5. The following table gives the ratio of the strength of the magnetic field due to a short magnet at points at varying distances from either pole of the magnet, to the strength of the earth's field. Find how the field varies with the distance (plot the logarithm of the inverse ratio against the logarithm of the distance) : Ratio of field to that of Earth. Distance from either Pole. 3-63 20 cm. I-OO 30 cm. 0-454 40 cm. 0-240 50 cm. 0-125 60 cm. 0-08 1 6 70 cm. 6. The table below gives the electrical resistance of 1000 yards of platinoid wire of different diameters. Draw a logarithmic graph connecting the resistance and diameter ; (a) what conclusion do you deduce from it, (b) what is the resistance of 1000 yards of platinoid wire of I mm. diameter ? Diameter in mm. - 1-626 1-219 0-914 0-711 o-559 o-457 Resistance in ohms per\ looo yards length / 180-3 320-6 870-0 942-2 1026 2280 A MODERN ARITHMETIC 7. To minimize the risk of fire due to electric lighting in a house, a piece of wire of fusible tin is fixed in a convenient position and the current passes through this ; but if the current becomes too strong, the wire fuses, and the current is stopped. Wires of different thicknesses are used according to the strength of the greatest current which can pass with safety. The following table gives the thicknesses and corresponding currents : Thickness in mm. 0-711 o-559 o-457 0-376 0-315 0-274 0-234 0-193 Current in amperes 77 5-35 3-96 2-95 2-26 1-84 1-44 1-08 Draw a graph connecting the logarithms of the thickness and current, (a) Deduce an approximate connexion between the thick- ness and current ; (^) find the thickness of the fuse if the current is to be not greater than 6 amperes ; (c) what current will just fuse a wire of 0-5 mm. diameter? 61. COMPOUND INTEREST. In the examples on Simple Interest, dealt with in 48, the principal is supposed to remain constant, so that in equal intervals of time equal amounts of interest accrue. In most actual cases, however, the interest is compound, and is added to the principal at the end of fixed intervals of time, seldom greater than a year. The following examples illustrate the more usual methods of procedure in different cases where the interest is compound : EXAMPLE i. What is the Compound Interest on ,536. i2S. in $\yrs. at 4 %per annum^ the interest being payable yearly t ^536. 1 2*. =,536-6. The interest on this sum for the first year is at the rate of 4 on every ^100 ; it is therefore T ^ of ^536.6 = ^21-464. This interest is added to the principal, and the sum serves as the Principal for the second year. COMPOUND INTEREST 415 Hence, the principal for the second year is .536-6 + 2 1-464 = ^558-064. The interest for the second year = ^55 8 - 6 4 x TTJ = ^22-32256. The sum of the interest for the second year and the principal for the second year serve as the Principal for the third year. Hence, the principal for the third year = ^558-064 + ^22.32256 = ,580.38656. The interest for the third year = ^23-2154624. The principal for the next half year = ^"603-6020224. The interest for the half year is \ x T - F of the principal, and .'. = ,1 2-072040448. Hence, the final amount is ^603.6020224 + ^12.072040448 = ^615.674062848; and on subtracting the original principal .536-6 from this amount, the total interest is obtained. .'. the interest = ^79-074062848 Principal for first Interest The method of calculation in its more common form is as follows : year = 5 36-6. - 21-464 Prin Inter rincipal for second nterest Principal for third Interest ,, Principal for last half Interest Final amount Original principal Interest (multiply by 0-04). (add the 4% interest). = 22-3226 (work only to nearest figure in 4th decimal place). = 580-3866. = 23-2155- = 603-6021. = 12-0720. = 6i5' 6 74i- (subtract the initial principal) = 79- 416 A MODERN ARITHMETIC In this example, the amount at the end of a full year = (1-04) times the principal at the beginning of the year; and the amount at the end of the final half year is (1-02) times the principal at the beginning of the year ; accordingly, the final amount = ;(5 36-6) x (i-o4) 3 x (1-02). It is well to put down the expression for the amount in this form even if the method of evalution be as in the example above. EXAMPLE 2. Find the Compound Interest on 610. 6s. in 4 yrs. at 2 1 % per annum, the interest being payable quarterly. If the rate be 2 J % per annum, then in \ year the interest on ^,100 will be J of 2-5 = ^0-625, and the interest on i will be ^0-00625, so that the amount at the end of any quarter = (principal at the beginning of the quarter) multiplied by 1-00625. The amount (k) at the end of 4 years, or 16 quarters, = ^(610-3) x(i-oo625) 16 . Employing logarithms, we obtain log (A) = log 6 1 0-3 + 1 6 log 1-00625 = 2-7855434+ 16(0-0027060) = 2-7855434 + 0-0432960 = 2-8288394; .', A = 674-279. The Interest = ,-674-279 - ^610-3 = ^63-979 = 63. igs. id. N.B. Four-figure logarithms are of little use in the evalua- tions of examples on Compound Interest ; seven-figure tables should be employed. EXAMPLE 3. Find the Compound Interest on 736. 4*. in 4 yrs. at 3 % per annum. Frequently it may be convenient to draw up an interest table giving the amounts at the end of various years, the origina principal being supposed to be i. Thus, in the presen' example, COMPOUND INTEREST 417 i at the end of i year becomes 1-03, ,, 2 years ,, 1-0609 (add 0-03 times 1-03), 3 1-092727 (add 0-031827 i.e. 0-03 times 1-0609), 4 .> 1-125509 (add 0-032782), true to the 6th decimal place, though this may not always be accurate enough. The Compound Interest on ^736. 4^ is therefore 0-125^ = ^92-400 = 92. 8s. _736-2 87-8563 3-7653 o-753i 251 92-3998 EXAMPLE 4. What sum will amount to ^1389. 35*. in 3 yrs. at 5 % per annum, Compound Interest, payable yearly ? i under these conditions will amount to (i-o$) 9 . (i'0$)* has resulted from the investment of ^i, and ^"1389. 3^. from the investment of ^(1389-15) -f (i-o5) 3 . Therefore the sum required (i 389-15) -7-1-157625 In finding both the rate per cent, and the time, logarithms are generally necessary. EXAMPLE 5. At what rate per cent, per annum will 1100 amount to 1202, i$s. gd. in 3 yrs., the interest being compound a?id payable half-yearly ? If 2 r be the nominal rate per annum, ;ioo becomes (ioo + r) at the end of one half year. Hence, six half years. J.M.A. 2 D 4I g A MODERN ARITHMETIC /ioo + A 6 _ 1202-7875 \ 100 / I 100 ioo + r 100 \ i ioo '//Tlbn = V( ni_ , The right-hand expression is then evaluated by logarithms. log ( IO ioo r ) == i( 1 g I202 '7 8 75- 1 g IIO ) = 1(3-0801889-3-0413927) = 1(0-0387962) = 0-0064660 ioo whence r=i-$ and 2^ = 3. The nominal rate per cent, is 3. EXAMPLE 6. In what time will ^1400 amount to ,2 1 7 1. \is. 2d. at 5% per annum, Compound Interest, payable yearly ? If t represent the number of years, ^(i4oo)(i-o5) f = x '2i7[. ijs. 2^ = ^2171-858. Employ logarithms. - 3-1461280 log 1-05 0-0211893 0-19070^4 = ~~- = 9 approx., 0-0211893 whence the time is 9 years. EXERCISES LXXXVII. a. Mental or Oral. Give expressions for the following amounts, the interest being compound and payable yearly : 1. ^1000 in 3 yrs., at 4 % per annum. 2. ^500 in 3 yrs., at 2 %. 3. ,500 in 2 yrs., at 3 %. COMPOUND INTEREST 419 Give expressions for the following amounts, the interest being compound and payable yearly : 4. ,1200 in 4 yrs., at 2^%. 5. ^800 in 5 yrs., at 3 %. 6. ^800 in 3 yrs., at 5 %. 7. ^900 in 6 yrs., at 4 %. 8. ^750 in 6 yrs., at 4 %. 9. ^100 in 4^ yrs., at 6 %. 10. 400 in 10 yrs., at 4 %. 11. ^400 in 10 yrs., at 3^ %. 12. ^500 in 2^ yrs., at 5 %. 13. ^750 in 3^ yrs., at 6 %. 14. ^1000 in 4^ yrs., at 4 %. 15. ,1000 in 4^ yrs., at 4^ %. If the nominal rate per cent, per annum be 4, but the interest be payable quarterly, then the rate per cent, per quarter is i, and i becomes ^"(i-oi) 4 at the end of a year; but (i-oi) 4 = 1-04060401, so that 4 % per annum nominal rate, the interest being payable quarterly, is the same as 4-060401, effective rate, payable yearly. EXEECISES LXXXVII. b. Mental or Oral. Give expressions for the interest in the following cases : Compound Interest on 1. ,500 in 3 yrs., at 4 %, payable half-yearly. 2. ,500 in 3^ yrs., at 4 %, payable half-yearly. 3. ,500 in 3^ yrs., at 4 %, payable half-yearly. 4. ,1000 in i yr., at 6 %, payable monthly. 5. 200 in 9 mo., at 3 %, payable quarterly. 6. ^1000 in 3 yrs. 4 mo., at 3 %, payable three times a year. 7. ^1200 in 2| yrs., at 6 %, payable monthly. 8. i in 9 mo., at 4 %, payable quarterly. 4 2o A MODERN ARITHMETIC EXERCISES LXXXVII. c. 1. Find the Compound Interest in the following cases : (a) ,1500 for 2 yrs., at 5 % per annum, payable yearly. (b) %o. 7s. 6d. for 4 yrs., at 4 %, (c) ,318. 15.?. for 2 yrs., at 3 %, (d) ^506. 8j. for 3 yrs., at 2^ %, (e) 700 for 2| yrs., at 4 %, (/) ^1986. I2J. for 2 yrs., at 3*- %, J - 6 ^- for 5 Y rs -> at 2 I %> -r- for 3 yrs. 3 mo -> at 3 %> (0 ^38,626. 6^. 6rtT. for 5^ yrs., at 5 %, 2. Find the effective rate, when the nominal rate per annum is as follows : (a) 4 %, payable half-yearly. () 4 %, payable monthly, (c) 6 %, payable monthly. (</) 6 %, payable half-yearly. 3. Draw up a set of Interest tables, showing the sums of money to which ,1 would amount in i, 2 ... 10 yrs., at (a) i %, (b) ii %, (c) 2 %, (rf) 2* %, (') 3 %, (/) 4 %, Or) 5 %, Compound Interest, payable yearly. State the answers true to the 7th decimal place. 4. Employ the tables drawn up in Ex. 3, to calculate the Interest on (a) ,350 in 2 yrs., at 4 % per annum, payable half-yearly. (b) ^440. 75. 6d. in i yrs., at 5 %, payable quarterly. (c) ;i53- I2J. in 3 yrs., at 5 %, payable half-yearly. (d) ,1080 in 4 yrs., at 2 %, payable half-yearly. (e) 2356. I2J. in 2 yrs., at 10 %, payable quarterly. (/) -^75 in 5 yrs., at 4 %, payable half-yearly. (g) ^3 m 2 Y rs - 3 mo -> at 4 % payable quarterly. COMPOUND INTEREST 421 EXERCISES LXXXVII. d. 1. In what time will ,306 amount to .372. js. gd., at 4 % per annum, compound interest, payable yearly ? 2. The sum of ^5200 was invested by the trustees of an orphan when he was 13 yrs. old. When the orphan became of age (21) the sum had amounted to ,6597. 4$. gd. What was the rate of interest, payable yearly ? 3. At what rate per cent, per annum, compound interest, payable half-yearly, will .333. 6j-. 8^. amount to ,375. 7s. gd. in 3 yrs.? 4. Find the sum of money which will amount to ^1292. 5-y. jd. in 3 yrs., at 2^ % per annum, compound interest, payable yearly. 5. What sum of money, invested at 2 % per annum, compound interest, at the birth of a boy, would amount to ,2000 when the boy attained the age of 25 years ? 6. In what time will ^1300 amount to .1436. os. qd., at 4 % per annum, compound interest, payable quarterly? 7. Employ logarithmic tables to find the effective rate of interest, if the nominal rate be i % per annum, but payable at equal periods. (a) 10 times a year, (b} 50 times a year, (c} 100 times a year. N.B. Note that the rates are all very nearly the same. The answers approximate to what the effective rate would be, if the turn over was immediate. 8. Employ logarithms to calculate what the effective rate would be if the nominal rate was 5 %, payable at the same intervals as in Ex.7. The Compound Interest Law. Geometric Progression. If a sum of money, say ^65, be put out at Compound Interest for example 5 per cent, payable yearly the amounts at the end of various years would be as follows : 422 A MODERN ARITHMETIC Amount ^65 (originally) becomes at the end of the ist yr. 2nd yr. 3rd yr. 4th yr. 65(1-05), 65(1-05)*, ^65(1-05)8, ^65(1-05)* etc, where it is seen that the amount at the end of any year is equal to the amount at the end of the preceding year multi- plied by a constant factor, 1-05. A series of terms of this kind is said to constitute a Geometric Progression. The following examples illustrate the various processes employed in connexion with such series : EXAMPLE i. Find the nth term of the series in geometric progression, of which the first five terms are 3, 6, 12, 24, 48. In this series the ist term is 3, 2nd 3x2, 3rd ,, 6 x 2 or 3 x 2 2 , 4th 12 x 2 or 3 x 2 3 and 5th 24 x 2 or 3 x 2 4 , and so on generally; the loth for example would be 3 x 2 10 " 1 . th 3 x 2"- 1 . EXAMPLE 2. A person possesses an annuity of 4% a year, for 10 years, payable quarterly, the first payment being made in i quarters time. What is the present value of such an annuity, at Compound Interest, reckoned at 3 % per annum, and also payable quarterly ? Here, a sum of money is to be found which would be the fair equivalent of the 40 payments. But, i paid now would amount to ,( 1-0075) in 3 months' time; and a payment of i in 3 months' time would be the equivalent of now: while 1-0075 a payment of i in 6 months' time would be the equivalent of ( ) now ; and COMPOUND INTEREST 423 the present value in of the 40 payments of 12 would be 12 12 12 12 (1-0075)2 + (1-0075)3 + ' ' -(i-oo75) = Say)) constituting a Geometric Progression, with as the con- stant ratio or factor. 1-007 5 Multiply by - , that is, by the constant factor or ratio. 1-0075 Then 12 12 . 12 12 T (1-0075)2 (1.0075)'" "(I-0075) 40 (i-<>75) 41 ' 1.00.75' Subtract this from the series for P. 12 12 _ P 1-0075 ( I>00 75) 41 i '0075* But = 1-0791813 -40(0-0032451)= 1-0791813 - 0-129804 = 0-949377 = log 8-89974; = 8-89974; and (1-0075)^0 12-8-89974 3-10026 .. from (i) P = = 413-368. 0-0075 0-0075 The present value is .*. ^"413. 7-v. ^d. EXAMPLE 3. A man puts by the sum of ^50 every year at 3 / Compound Interest, payable yearly ; what will be the accumulated sum iust before the 2 ist payment? The first ^50 accumulates interest for 20 yrs., and there- fore eventually amounts to ^5o(i-o3) 20 . The second ^50 accumulates interest for 19 yrs., and therefore amounts to oi-o 19 . 4 2 4 A MODERN ARITHMETIC The last $o accumulates interest for i yr., and amounts to s( l '3)' The accumulated sum in is therefore 5o(i-o 3 ) + 5o(i-o3) 2 +...5o(i-o3) 20 = S(say). Multiply by (1-03), the constant ratio, and subtract, as in the last example : ' 5(i-3) 21 -5(i-3) = 8(0-03). Employ logarithms, and (i-O3) 21 is found to be 1-86029. Whence, S = 5^M3^9}. 0-03 The accumulated sum = ^1383. i6s. od. approx. EXAMPLE 4. The population of the City of London was 37702 in 1891 and 26923 in 1901. If the rate of change per 1000 per year remained constant, what would you expect the population to be in 1906 ? Suppose the population at the end of a year = population at the beginning of the year multiplied by a factor x. Then the population 26923 = (37702).* 10 ; the estimated population in 1906 = (26923)^ (6923 - (26923) . ( 26923 ) '137702J 5 'l37702 (37702)* = P (say). Employ logarithms. log P = f log 26923 - 1 log 37702 = 6-6451852- 2~288l822 = log 22751. Estimated population is 22751. EXAMPLE 5. Find the sum of an infinitely large number of terms of the series 2 + f + + / T + . . . . Let s be the sum. 5 ~ 2 + + f + TT + for an infinitely great number of terms. Multiply by the constant factor 1. COMPOUND INTEREST 425 ' = f 4-f 4 Y V+ f r an infinitely great number of terms ; but the end terms become indefinitely small, whence on subtraction f i c _ ,, . li .? 2 1 ~ a EXAMPLE 6. Convert 0-365 zVzfo vulgar fraction. 0-365 = 0-365 4- 0-000365 4- 0-000000365 4- 0-000000000365 4- . . . , 0-365 = 0-00036 5 4- o-oooooo 3 6? 1000 4- 0-000000000365 4- . . . , subtract (o-36c;)(i ^ = 0-365; 0-365=^5. 999 EXERCISES LXXXVII. e. 1. Express the following recurring decimals as series in Geometric Progression, and hence calculate the decimals as equivalent vulgar fractions : (a) 0-4; (b) 0-123; (c) 2-236. 2. Calculate the present value of (a) ^109. 5^. 5^/., due 3 years hence, at 3 % per annum, compound interest, payable yearly. W .763. 7s. 8^/., due 20 years hence, at 4 % per annum, compound interest, payable yearly. 3. Verify the entries in the table at the end of the book of present values in the following cases : (a} By direct calculation, for (i) ;i in 3 yrs., at 3 %, payable yearly. (ii) 1 in 4 yrs., at 5 %, (iii) .1 in 6 yrs., at 4 %, 426 A MODERN ARITHMETIC (b] By the use of logarithms : (i) ^i in 15 yrs., at 2| %, payable yearly, (ii) i in 20 yrs., at 3^ %, (iii) ji in 30 yrs., at 4 %, (c) Assuming the truth of the entries, as verified in prove the truth of the entries corresponding to (i) i in 16 yrs., at i\ % . (ii) i in 21 yrs., at s % . (iii) 1 in 31 yrs., at 4 % . 4. Evaluate the present value of the following annuities, the first payment being made at the end of the first year : (a) 100 for 10 yrs., reckoning compound interest at 3 % per annum. 14 4% 7 2% (d) 120 16 3% 5. A steel ball, allowed to bounce vertically, rises to heights which gradually decrease in such a way that the height in any one bounce is 80 % of the height in the preceding bounce. What is the height of the 5th bounce, if the ball be first thrown up to a height of 6 ft. ? Find also the total distance moved through before finally stopping. 6. A ball bounces as in Ex. 5, but so that the height of any bounce is a certain multiple (x) of the height of the bounce immediately before. It rises to a height of 2-4 metres when first thrown up, and moves through a total distance of 16 metres before coming to rest. Find x. 7. When a rope is coiled round a post, a person exerting a small pull on one end can withstand a much greater pull at the other end, the pull decreasing, according to the compound interest law, with the number of turns of the rope. If, when the rope is coiled once round the post, a pull, equal to the weight of i lb., can withstand a pull of 16 lb., find the pull COMPOUND INTEREST 427 necessary to withstand a pull of 16 tons, if the rope be coiled 4 times round the post. State the answer true to 3 significant figures. 8. When light passes through glass, a certain fraction of the light falling upon the glass passes through ; this fraction varies with the thickness of the glass according to the compound interest law. A certain piece of glass, i mm. thick, allows 95 % of the light to pass through ; if the thickness be doubled, only (0-95)2 of the light passes through. How much of the light will be transmitted by a plate 2-5 cm. thick ? 9. If^i be put by yearly at 3%, compound interest, payable yearly, then n contributions will amount to directly the last contribution is paid. Find the amount when n is successively (a) 3, (U) 4, (c) 5. 10. Verify (using logarithms) the entries in the table at the end of the book, giving the amounts in different times of yearly contri- butions of ;i, in the case where the number of contributions is 20 and the rates of interest (a) 2-5 %, (V) 4 %, (c) 5 %. 11. A colliery is estimated to yield an income for 30 yrs., and the purchase money is .54,930. What sum of money should be put annually on one side, so as to redeem the purchase money at the end of 30 yrs., reckoning compound interest at 3 % per annum, payable yearly ? 12. A person put aside 20 yearly for his son, beginning at birth. What amount was due to the son on his 2ist birthday (including the 22nd instalment) ? Assume that the money accumu- lated at compound interest, at 4 % per annum. 13. What annuity should be paid for 16 yrs. for ^100 paid now, the first payment to be made in i year's time, and afterwards in yearly payments, reckoning compound interest at 2-5 % per annum ? 428 A MODERN ARITHMETIC 14. A man, when 30 years old, puts by ^40. 4$. every year, which sum accumulates at 4 % compound interest. What annuity could he buy with the accumulated proceeds when 55 yrs. old, if, at that age, he can obtain an annuity of S on payment of 100? 15. A loan of ^1000 can be repaid in 5 years, together with its interest at 4% per annum convertible half-yearly, by 10 equal half-yearly payments of i 1 1. 6s. b\d. (or more exactly i 1 1-32653). Construct and complete a table as indicated below, showing the amounts of principal and interest contained respectively in each half-yearly repayment : Half-year. No. Outstanding Principal in ,. Interest for Half-year in ,. Principal in repaid in each Half- yearly Repayment (i.e. difference between Interest and Repayment). I 1000 20 91-327 (i.e. 111-327-20) 2 1000-91-327 (i.e. 908-673) I8-I74 93-153 (i.e. 111-327- 18-174) 3 p 4 p p 5 p p 6 p . ? 7 p ? 8 p ? 9 p ? 10 p p 109-144 Total i ooo-oo COMPOUND INTEREST 429 16. A certain insurance company will pay ^100 at the end of the given term of years for a single or annual premium as given by the following table : PREMIUM. To insure the return of ,100 at the expiration of a given term of years. Terms of Years. Annual Premium. Single Premium 10 s. d. 9 3 8 s. d. 78 6 6 20 3 18 5 58 5 8 30 244 43 7 5 40 i 8 o 32 5 5 50 o 18 9 24 o 3 60 12 II 17 17 4 70 092 13 5 ii 99 037 5 12 ii Draw a graph showing the change of premium with the time. Show that the single premiums are not calculated strictly accordingly to the Com- pound Interest Law. EXERCISES LXXXVII. f. Practical. [Apparatus : Weight. String. Support. Scale. Glass marble. Graduated paper circle^ 1. Suspend a heavy weight by two strings of the same length (about 6 ft. or more) to two hooks about 2 ft. apart. Fix a pointer to the weight and arrange a metre scale beneath, so that when the weight is allowed to swing, the pointer moves in front of the scale. Note the extent of, say, 50 successive swings on the same side ; determine the law according to which the swings die away. (This is best done by a graph of the logarithms of the successive swings.) 2. Allow a glass marble to fall upon a stone slab ; it rebounds ; alter the distance through which it falls until the height of the 430 A MODERN ARITHMETIC rebound is a definite, convenient amount (say 2 ft.)- Find the ratio between the height of the rebound and the original fall. Similarly, obtain the ratio of the height of the second rebound to the original fall, and so on. Arrange these ratios in a series, and determine its character. 3. Attach a weight with horizontal pointer to a support by means of a wire or fine thread ; underneath the pointer fix a horizontal graduated circle ; as the thread twists and untwists the arrangement will oscillate. Measure the 'sizes' of the successive oscillations, and determine the law according to which they decrease. EXERCISES LXXXVII. g. Graphic. 1. In pumping out the air from a vessel by means of an air- pump of the Hawksbee type, the amount of air in the vessel decreases with the number of strokes according to the compound interest law. The initial amount of air is 0-12 gr., and after the first troke the amount is only 0-096 gr. Draw a graph showing how (a) the amount changes with the number of strokes ; (/;) the logarithm of the amount changes with the logarithm of the number of strokes. Find (c) when the amount present will only be o-ooi gr. 2. Use the tables at the end of the book to draw a graph showing the sum to which yearly payments of ,1 will amount in different times at (a) i\%; (V) 3 % ; (c) 4 % ; (d) 5 %, com- pound interest payable yearly. Employ the graphs to find : (e) The time in which the yearly payment of ;i at 3 % will amount to ,50. (/) The amount after 18 yearly payments of i at 3 %. (g) The time when the amount at 5 % is twice as much as the amount at 2^ %. COMPOUND INTEREST 431 3. In an electrical experiment, the discharge from a condenser through a galvanometer gave an initial throw or deflexion of 330 divisions ; the deflexion gradually died away according to the compound interest law and the I5th deflexion was only 120 divisions. Draw a graph connecting the extent of the deflexion with the order of the deflexion (use logarithms). What was the amplitude of the loth deflexion ? 4. Draw a graph showing the present value of an annuity of i a year at (a) 2% % ; (b) 3 % ; (c) 4 % ; (</) 5 % Use the tables at the end of the book. Employ your graph to calculate (e) the value of an annuity of i for 12 years at 2\ % ; (/) the duration of an annuity of i, bought for ^16, interest 2^ %. 5. Draw a graph showing the present value of an annuity of ^i a year for n years as the rate per cent, changes from Rate, - 2i% 3% 3i% 4% 4i% 5% Present value, 9-5H 9-252 9-002 8-761 8-529 8-306 What is (a) the present value of the above annuity at 3^ % ; (b) the rate per cent, if the present value is 8 guineas ? 6. For how many years must a man put by a certain sum yearly in order to obtain afterwards an annuity of double that sum for the same number of years, the rate per cent, being (a) 4 %, (fy 5 % (Find the intersection of a graph showing the present value of i yearly for I, 2, 3, ... years, and a graph showing" the difference between the present value of 2. for i and 2 years, 2 and 4 years, 3 and 6 years and so on.) 7. Draw a graph showing how the present value of ^10 varies with the time, the rate of interest being 5 % ; on the same piece of paper and to the same scale draw also a graph showing how the present value of yearly payments of ^i varies with the time. Find the point of intersection of the two, and hence find how many payments of ,1 yearly are the equivalent of a payment of ,10 at the end of the time. 432 A MODERN ARITHMETIC 8. From a man A, a bank receives ^1000 on deposit at 2^ payable yearly. The bank, however, lends this ^1000 to B at 5 % payable yearly. The interest received by the bank can also yield 5 % payable yearly. Find when the bank can repay the 1000 to A, together with the accumulated interest, and still have the ^1000 to its credit on account of the loan to B. (Draw a graph showing the amount of ^1000 at various times at 2\ %, also a graph showing the amount on account of yearly payments of ^50 at 5 %. Find the point of intersection.) 9. The table below gives the number of years' purchase of a lease as the duration of the lease changes, if the rate of interest be assumed at 10 %. Find (a) the years' purchase if the duration be 1 1 years, (b) the duration if the years' purchase be 8. Years' duration 24 20 16 12 8 4 i Years' purchase 9 8-5 7-8 6-8 5-34 3-2 0-91 62. THE AREA OF A TRIANGLE. It has already been seen that the area of a triangle is equal to half the product of the length of its base and its altitude. The area can also be calculated when the lengths of the three sides are given. EXAMPLE i. Find the area of a triangle ABC of which the sides AB, BC, CA are 26, 42 and 40 units of length respectively. In Fig. 78 let AD be at right angles to BC, O the middle point of BC. But AC 2 = CD 2 + AD 2 ; .'. AC 2 -AB 2 = CD 2 -BD 2 . (CD-BD)(CD + BD) = 2(OD)(CB). THE AREA OF A TRIANGLE CD 2 - BD 2 AC 2 - AB 2 433 2BC 2 BC Hence, OD and BD = BO - DO. Since AD 2 = AB 2 - BD 2 , AD is now known, and the area of the triangle = J {AD x BC}. A A B D O FIG. 78. 32 FIG. 79. In this example CD' 2 - DB 2 = 4o 2 - 26 2 = 66 x 14 ; and 2BC.OD = 2 ( 4 2)(OD); AD 2 = 26 2 - io 2 = 36 x 16 ; .'. AD = 24, and the area = 1x24x42 = 504 units. In a great many cases application is made of the fact, that the areas of similar triangles (i.e. triangles of the same shape) are proportional to the squares of their corresponding sides. (Exer- cises LXXIX. b.) EXAMPLE 2. ABC is a triangle, and the portions ADE, CFG, BHI are cutoff; DE being parallel to BC. FG AB. IH AC. If AB is 1 in., BC 6 in., CA 8 in., and if AD is i in., Bl 2 in., CF i in., what fraction is the area of the remaining portion DEFGHI of the whole area ABC ? Since AD=i in. and AB=y in. B <-- --G"-- and the triangles ADE, ABC are of FlG - 8o - i 2 the same shape, the area of ADE = -3 of the area of ABC. 7 J.M.A. 2 E 434 A MODERN ARITHMETIC 2 2 Similarly, the area of BIH =^ of the area of ABC. 7 CFG = |2 The area of the three triangles together = (-3 + -2 + 02) ^ the area of ABC. V7 7 Hence DEFGHI has an area of i -~-^ of the whole triangle of the triangle ABC. 3136 EXEECISES LXXXVIII. ABC is a triangle, D the foot of the perpendicular from A to BC, O the middle point of BC. Calculate (a) OD, (b) AD, (c) the area of the triangle ABC in the following cases (the lengths are given in inches) : 1. AB = 65, BC- 57, CA =68. 2. 55 = 156, 55 = 176, 55 = 68. 3. 55 = 34, 55 == 42, 55 = 20. 4. = 5 1 , 55 == 55, 55 = 26. 5. 55 = 51. = 77, ,5 -40. 6. ,5 = 20, ,5 = 21, 51 = 13- 7. ,5 - 100, 55 = 9 1 , 5, =61. 8. = 6 5> 55 36, 55 = 61. 9. AB = b, BC 5 = , ( DA = < 10. 11. The area of a triangle ABC is 5000 sq. in., the base BC is loo in., the side AB is 125 in. ; calculate the length of the side AC to the nearest inch. 12. If in the preceding exercise the area=273o sq. in., BC = 9i in., AB = 6i in., find the value of AC. 13. Let the sides of a triangle ABC be represented by BC = <2, CA = <, AB = r, and let AD be the perpendicular drawn from A to BC, and O the middle point of BC. THE AREA OF A TRIANGLE 435 ffi * C^ [ The symbol -means that the smaller SnOW that (l) OD = -- , of the two quantities is to be taken 2<2 from the greater.} Also show that the last expression 16 ' This last formula is suitable for logarithmic computation. 14. Apply the expression developed in the preceding exercise to the calculation (true to four significant figures) of the area of the triangles, the sides of which are respectively : (a) 203, 218, 324. (b) 96-02, 89-3, 52. (c} 825-4, 1018, 918-2. (^0 937, 135, MI- 0) 671, 528, 609. (/) 42-3, 59-3, 37-6. Cf) 972, 97i, 63-2. (h) 361, 258, 372. (z) 1096, 1342, 1723. 0) 1562, 437, 1485. (k) 2374, 1836, 1972. (/) 692, 347, 560. 15. ABC is a triangle, D and E two points on AB and AC respec- tively, such that AD-rAB = AE4-AC. Find the ratio of the portions into which the triangle is divided by DE when isMl, (*)i, W|(rf)3. Mi. </)f, (^)i- 16. ABC is a triangle, such that BC= loin., CA= 11 in., AB=I2 in. DEF are points on BC, CA, AB respectively, such that BD = 3 in., CE = 4 in., AF = 5 in. Calculate the areas of the triangles (a) BDF, (b) CDE, (c) AFE, (d) DEF in terms of that of the triangle ABC. 17. In a triangle ABC, BC=i3 in., CA=i5 in., AB = i6 in. DE is a line parallel to BC, dividing the triangle ABC into two equal parts and meeting AB, AC, at D and E respectively : find the length of AD true to 3 significant figures. 43 6 A MODERN ARITHMETIC 63. BILLS OF EXCHANGE AND BANKER'S DISCOUNT. Debts are frequently cancelled by means of Bills of Exchange. A bill of exchange may be drawn up in the following fashion : London, \$th March, 1905. ^375- I0 ^ Twenty-five days after date pay to the order of Mr. A. B. the sum of Three Hundred and Seventy-five Pounds Ten Shillings, value received. C. D. Mr. E. F., Such a bill would be drawn by C. D. and accepted by E. F., who would write his acceptance across the bill, while A. B. is the payee. Once the bill has been accepted, it becomes negotiable, and may pass through several hands before being presented finally at a Bank. If presented before the date of maturity, together with three days of Grace (i.e. i3th March + 25 days + 3 days), i.e. April loth, the banker deducts an amount of Discount, calculated in the same manner as ordinary interest on the number of days, through which the bill has still to run. Thus, in the above example, if the rate of discounting be 3%, and if the bill be discounted on March 3ist, that is 10 days before April 10, the banker deducts The Banker's Discount is therefore 6s. 2d. BILLS OF EXCHANGE AND BANKER'S DISCOUNT 437 EXERCISES LXXXIX. Find the Banker's discount on the following bills (3 days' Grace to be included) : Bill for Maturing on Date of Discounting. Rate /.. 1. .890. \2s. 6d. May 4, 1905 April 3, 1905 3 2. ^1896. ios. 3 mo. from April 17, June 18, 1905 3 1905 3. ,250. 17^. 6d. June 10, 1906 June r, 1906 2j 4. .3864 6 mo. from April 3, June 10, 1906 4 1906 5. ^7876. 5J. 6 mo. from Feb. 2, May 8, 1906 3i 1906 6. ^i75o March 10, 1904 Feb. 3, 1904 4 7. ^387. 17 s. 6d. March 12, 1904 Feb. 1 6, 1904 4* 8. &7$o 6 wks. from Ap. 3, Mar. 28, 1904 3i 1904 9. ^1568 Sept. 15, 1905 Sept. i, 1905 2f 10. ,18,763 Oct. 14, 1905 Sept. 29, 1905 4 11. ,1266. i$s. 4d. Dec. u, 1905 Dec. i, 1905 3* 12. ^836. I7J. 6d. Dec. 23, 1905 Dec. 10, 1905 2| 13. ,9451. i2s. 6d. Nov. 26, 1904 Nov. 23, 1904 3 14. ^8615 4 wks. from Ap. 15, April i, 1.905 2i 1905 15. ,1020. i is. yd. July 28, 1906 July 3, 1906 3 16. ^1237. Ss. 04. Aug. 3, 1906 July I, 1906 3i 17. ^1193. is. 6d. 70 days from July 2, Sept. i, 1906 4 1906 18. ^8765 25 days from May 2, May 15, 1905 2} 1906 19. ^9876 4 mo. from Dec. 4, Jan. 24, 1905 4i 1904 20. ! ,219. I2J. 3d 5 mo. from Aug. i, Nov. i, 1904 3 1904 438 A MODERN ARITHMETIC Find the terms missing from the following tables : 21. Bill. Date of Maturing. Date of Discounting. Banker's Discount. Rate % ^376 April 3, 1906 ? 3. 155. 2\d. 5 22. ? July 28, 1905 July 10, 1905 l. $s. 2\d. 2i 23. ? May 30, 1904 Mar. 21, 1904 & is*- 3i 24. 1200 ? June I, 1905 2. 19.9. g\d. 2| 25. 3750 Sept. 17, 1905 ? 2. us. gd. ?J 26. ^"1872. ios. Aug. 28, 1 906 Aug. i, 1906 ^4. 135. 6d. p 27. ^4015 Oct. 10, 1905 Aug. 31, 1905 ^14. y. 9fc</. p 28. 456. 5-r- July 28, 1905 ? o. iSs. gd. 2* 29. 608. 6s. 8rf. ? Feb. 26, 1905 i. &. 3i 30. 406. i3j. 4d. Jan. 26, 1905 ? ^o. 15*. 4i 31. 501. ijs. 6d. May 22, 1905 May I, 1905 o. i6s. 6d. P 64. CALCULATION OF AREAS (continued}. Conical Surfaces. Suppose AOB represents a sector of a circle (i.e. an area bounded by the circular arc AB and the two radii OA and OB). A FIG. 81. Its area = JOA (length of arc AB). Wrap up the surface so that AB becomes the circumference of a circle and AO coincides with BO. The surface now becomes Conical, and it is clear that CALCULATION OF AREAS 439 The curved surface of the cone = area of the sector, and /. , , , , = i (slant edge ) (circumference of base). .'. If h be the distance D between the apex O and the centre c of the base (Fig. 81), r= the radius of the base and /= the slant edge. The area of a cone = J/x 2Trr-=irlr In. dealing with frusta of a cone the principle of similar figures can often be used. (The frustum of a cone is the part between the base and a plane parallel to the base.) If a frustum of a cone be unwrapped, it will be seen that the curved surface is the difference between the areas of two similar circular sectors (Fig. 82). If n be the number of degrees formed by the two bounding radii of the sector and L x and L 2 the radii, FIG. 82. the area of the sector = 300 300 7rL 2 2 (see p. 333) = (difference in radii) (mean of the two arcs), whence the curved surface of the frustum of a cone = TT (slant edge) x (mean of the diameters of the ends). 440 A MODERN ARITHMETIC EXERCISES XC. 1. A circle of 4 in. radius is divided into four equal sectors ; if one of these sectors be folded into the form of a cone, what is (a) the radius of its base and (b) its altitude ? 2. If, from a circle of 6 in. radius, a sector be formed containing an angle of 288, what will be (a) the altitude, (b) the radius of the base of the conical surface which can be formed from it ? 3. Find (i) the altitudes and (ii) radii of the bases of the conical surfaces which can be formed from the following sectors : (a) Radius of circle 15 cm., angle subtended at the centre 30 (b) 18 cm., 45 (c) 9 cm., 50 (d) 36 cm., . 1 60 (e) 14 cm., 209 (/) 13 cm., length of arc 31-41 cm. (/) 6 1 cm., 69-11 (h) 17 cm., 50-26 (z) 25 cm., 94-24' 0) A J. 4. A conical tent is to be 8 ft. high, the radius of its base 6 ft. ; what amount of canvas is needed for its construction (neglecting seams, etc.)? 5. An old-fashioned candle snuffer was in the form of a cone ; if its altitude was 3 in., radius of base i in., what was its outside surface ? 6. The bearing of a shaft is in the form of a cone, 5 in. long, radius 3 in. ; what is its total surface ? 7. Find the curved surface of the following cones : (a) Slant edge 10-125 m -> radius of base 8 in. (^) 15 ft- 5 ft. (c) 8ft. 7 ft. 00 9 ft- 6-7 ft. (e) 8ft. 4-6 ft. CALCULATION OF AREAS 441 8. A tin-plate basin is in the form of a frustum of a cone, with the diameters of the upper and lower rims 18 in. and 12 in. respectively, the slant edge is 5 in. ; how many square inches of tin-plate does it contain ? 9. A tin funnel is in the form of two frusta. The upper one has its upper rim 6 in. in diameter and lower rim 1-2 in. diameter ; the lower frustum has its upper rim 1-2 in. diameter and its lower rim 0-6 in. diameter ; the slant edges of the upper and lower frusta are respectively 5 in. and 4 in. What amount of tin-plate does the funnel contain ? 10. A tin can consists of (i) an upper cylindrical portion 4 in. long, 6 in. diameter, (2) a lower cylindrical portion 18 in. long, 12 in. diameter, (3) a connecting conical frustum, axial length 4 in., and (4) a circular base. What is the total surface of tin-plate used in its construction ? 11. Find the outside surface of a flower pot, of which the outer dimensions are : height 12 in., upper rim diameter 9 in., lower rim diameter 6 in., drainage hole in base i in. diameter. 12. Find the curved surface of the following conical frusta : Diameter of upper rim. Diameter of lower rim. Altitude. (a) i cm. 1 8 cm. 4 cm. w 13-5 in. 18-5 in. 6 in. w 20 in. 42 in. 60 in. w 3'4 in- 6-6 in. 3 in. to 100 in. 340 in. 22 in. Spherical Surfaces. Suppose the sphere in Fig. 83 to be divided by parallel planes into a number of thin zones, such as PQ. The surface of all these zones make up the whole surface of the sphere. But the zone PQ may be also regarded as a portion of a 442 A MODERN ARITHMETIC o Tlf^Y conical surface which touches the sphere, and with its apex, say, at T. .'. the area of the zone = PQx mean perimeter (see p. 439) = PQ x 2?rAP (PQ being very small). But since PQT is perpendicular to OP, O being the centre of the sphere, and since AP is perpen- dicular to OA the As OAP, PTA have the same shape, and AA' _ AT _ AP PQ ~ PT ~ OP' where A, A' are the centres of the circles through P and Q. .'. AA' x OP = AP x PQ, and the surface of the zone = 27TOP X AA'. Hence the surface of a zone of a sphere of radius r=2irrd, d being the distance between the two planes which bound the zone ; and the surface of the complete sphere = 2irr . 2r = 4irr 2 . FlG EXERCISES XCI. a. Find the surfaces of the spheres of which the diameters are : 1. 3- 1 1 in. 2. 12-3 in. 3. 2-02 cm. 4. 248 cm. 5. 30-2 cm. 6. 24-0 in. 7. i-i ft. 8. 50-2 cm. 9. 42 cm. Find the diameters of the spheres of which the surfaces are : 10. 5728-84 sq. cm. 11. 6734-61 sq. ft. 12. 81-7130 sq. in. 13. 333-293 sq. cm. 14. 2865-26 sq. ft. 15. 1269-23 sq. cm. 16. 725-835 sq. in. 17. 4584'35 sq. yd. CALCULATION OF AREAS 443 EXERCISES XCI. b. (TT may be taken as 3-1416.) 1. If the earth be regarded as a sphere, having a radius 4000 miles in length, what is its surface ? 2. What is the surface of a cricket ball, diameter 375 in. ? 3. The curved surface of a hemisphere is 10331-48 sq. cm. ; what is its radius? 4. A hollow brass sphere is 4 in. in diameter ; what is the area of its outside surface? What also is the mass of the sphere, if the thickness be o-oi in. (density of brass 8-816)? 5. If the surface of the earth (regarded as a sphere, radius 4000 miles) be supposed ruled by the 360 meridians or lines of longitude, and so divided into 360 equal parts, find in square miles the surface of the earth between longitudes (a) 30 E. and 30 W. (b) 25 6' E. and 16' E. (c) 140 E. and 90 E. (*) 76 W. and 15 E. (*) 25 10' W. and 45 20' E. (/) 126 10' W. and 10 W. Gr) 79 W. and 78 W. 6. The table below gives the distances from the centre of the earth (radius 4000 miles approx.) to the centres of the following parallels of latitude, while 2 3 i N. is the Tropic of Cancer \ ^^ bmmd the ^ 23^ b. Capricorn J 66^ N. is the Arctic circle. 66^ S. Antarctic circle. Latitude IO 20 23i 30 40 North Di stance = radius multiplied by 0-1736 0-3420 0-3987 0-5000 0-6428 Latitude 50 60 66i 70 80 North Distance = radius 'multiplied by 0-7660 0-8660 0-9171 Q-9397 0-9848 444 A MODERN ARITHMETIC Calculate in square miles to four significant figures the surface (a) each of the Frigid Zones. () Temperate Zones. (c) the Torrid Zone. (d) the Earth between latitudes 20 N. and 40 N. (e) 20 N. 40 S. (/) the portion between parallels 20 N. and 40 S. and longi- tudes 10 E. and 11 E. (g) a portion of a Frigid Zone between 80 E. and 85 30' E. (//) a portion of the Torrid Zone between 75 E. and 10 E. 7. The inside of a glass goblet is made in the form of a portion of a hemisphere of 5 in. radius, with the radius of the lower and smaller end 3 in. : what is its total inside curved surface ? 8. A boiler is in the form of a cylinder with hemispherical ends : its total length is 12 ft., its diameter 4 ft. ; what is its surface ? 9. What is the total surface of a hemisphere, of diameter 8-26 cm. ? 10. A hollow spherical shell is i in. thick and its outer diameter is 6 in. ; what is its total outside and inside surface ? 11. In Exercise 10, what would be the total surface if the shell were cut in halves and separated ? 12. If each square centimetre of each surface of a film of water be the seat of energy sufficient to raise i gram weight through a vertical distance of 0-7 mm., what vertical distance could i gr. be lifted through by utilizing the energy of a soap bubble 24 cm. in diameter ? 13. A cauldron is in the form of a hemisphere, its diameter is 6-2 ft. and thickness i in. ; find its weight approximately, the weight of i cub. ft. of the substance being 450 Ib. CALCULATION OF AREAS 445 EXERCISES XCI. c. Practical. [Apparatus : Sheets of metal or cardboard. Shears. Balance.} 1. Cut out a number of triangles from a piece of sheet 'metal or cardboard. Cut out also a square of, say, 5 cm. side. Calculate the areas of the triangles (a) by measuring the length of the sides, and then computing ; (b} by comparing the weights of the triangles with that of the square. 2. Draw a quadrant of a circle, radius 2 in. Beginning from one end, mark off a number of small arcs (AB may be taken as a typical one). Draw perpendiculars (such as ANJ from the ends to one of the bounding radii. Draw a graph by marking along one line Qx the length of the arcs and setting off perpendiculars equal to the perpendiculars found above. Find the area contained between O^r, Oy and the graph. Calculate the ratio of the area to that of the quadrant above. 3. Draw a circle upon squared paper, and determine its area. Find also the area of a square drawn upon the radius of the circle. Divide the former by the latter result, so as to determine the ratio between the area of a circle and the area of the square on the radius. 446 A MODERN ARITHMETIC 4. Draw a circle upon squared paper, taking as radius distance between 10 lines. Count the number of squares in the circle, estimating, as before (p. 127), the fractions of squares. Now calculate the area from the formula, area of circle = 7rxrad. 2 , the radius being 10. Thus verify practically the accuracy of the formula. It is difficult to determine experimentally the area of the surface of a sphere, but the following device provides a very approximate illustration of the rule for calculating the area from the radius : 5. Make or procure the apparatus shown in Fig. 85. To make the instru- ment, a narrow metal ring with a neck is obtained. Thin sheet india-rubber is fastened upon the ring, so as to form a hollow box with flexible ends. A square centimetre is drawn upon the india- rubber. A flat disc, which can be inflated into a sphere of the same diameter, is thus obtained. By means of a tube fastened upon the neck blow out the india- rubber until it takes the form of a ball. While it is stretched measure the area of the square marked upon it. It will be found approximately double what it was originally, thus indicating that the surface of a complete sphere has four times the area of the hemi- spherical section or flat part obtained by cutting it into halves. 6. That the surface area of a sphere is four times the area of a circle of the same diameter as the sphere can also be de- monstrated by weighing hemispherical shells of sheet brass and circular dies of the same material,* diameter and thickness as the sphere. If the hemispheres are fairly large, say three inches in diameter, two of them together weigh, within a fair degree of accuracy, four times as much as a disc. Here is an actual result : Mass of disc, .... 22-1 grams. Mass of two hemispheres, - - 88-8 grams. FIG. 85. * [Brass hemispheres and discs can be obtained from Messrs. Griffin and Sons, Ltd., Kingsway, London, W.C.] CALCULATION OF AREAS 447 7. The area of the surface of a sphere may also be found in the following way : Procure an old tennis ball. Measure its diameter and then cut off the flannel covering ; flatten each half section down on squared paper and find the area. Add the two to get the area of the whole surface. Hence prove that the area of the sphere is four times that of a circle of the same diameter. A fives ball may be similarly employed, the four coatings being cut off and their areas measured. 8. Weigh a thin hollow spherical shell of metal, measure its diameter (d\ also the thickness of the metal ; calculate and com- pare the weight of the shell with rrd 2 multiplied by the weight of unit area of the sheet metal. 65. CALCULATION OF VOLUMES (continued). Volumes of Cones and Pyramids. ABCDEFGH (Fig. 86) is a cube, O is the central point. The cube could be divided into 6 exactly equal parts by the planes OAB, OBC, and so on. Hence, in this particular case, the volume of a single pyramid, such as OABCD, is J(area of base) (altitude), being ^th that of the cube. The pyramid may be regarded as the sum of an exceedingly great number of square strips, such as abed (Fig. 87), and each strip may be supposed slipped along until the form of the pyramid is changed from Fig. 87 to Fig. 88, so that in this case also the volume=-g(area of base) (altitude). Reasoning in this fashion, it can be shown that the volume of any square pyramid = ^ (area of base)(altitude). A MODERN ARITHMETIC Again, the series of strips, such as abed (Fig. 88), might have been replaced by a series of other shaped strips of the sam< FIG. FIG. 90. area (Figs. 89 and 90). So that the volume of any cone or pyramid = \ (area of base) (altitude). Volumes of Frusta. EXAMPLE. Find the volume of the frustum of a cone, the radii of the plane ends being 7 inches and \ o inches respectively, and the height of the frustum 9 inches (Fig. 91). In this and in most cases of a similar character, the problem is best solved by the use of similar figures. If a cone be imagined cut across by a number of parallel planes all perpendicular to the axis, but at different distances from the apex of the cone ; the radius of any section is pro- portional to the distance of the section from the apex, and CALCULATION OF VOLUMES 449 the increase in the radius is proportional to the increase in the distance of the section from the apex. Thus, in this example, if ABCD (Fig. 92) be half a section of the frustum through the axis, O the position of the apex of the cone of which the frustum forms a part, and if CE be parallel to the axis OAB, v * ' vi ~ . /- V ' \ OA = CE AC~ED' 7><9 (AC)(CE) (BD-AC) = 21, 10-7 whence also OB = 2 1 + 9 = 30. The frustum is therefore the difference between two cones, one of height 30 inches and end radius 10 inches, the other of height 21 inches and end radius 7 inches. The volume of a cone of altitude 30 inches and end radius 10 inches is ^ir x io 2 x 30 cubic inches (i) O A B FIG. 92. The volume of a similar cone of altitude 21 and end radius 7 = (f^) 3 x ITT x io 2 x 30 cub. in., or = (yV) 3 x 3 x TT x io 2 x 30 cub. in. ; (ii) therefore subtracting (ii) from (i) the volume of the frustum ( / 7 \ 3 l7rx io 2 x 30 -{'-()}- = 6 "- Similarly, in the general case where the radii of the ends are R . r and the altitude h, The complete cone, of which the frustum may be regarded D as part, has an altitude = K T J.M.A. 2 F 45 c A MODERN ARITHMETIC R-r The volume of this complete cone = J?rR 2 . - - . h. The volume of the frustrum _ jf\l X 7TR 2 X This may also be written R~r and still more generally where S . s are the areas of the ends ; and in this form the expression is suitable for calculating the volume of a frustum (Fig. 93) of a pyramid or FIG. 93. of any conical body. Volumes of Spheres. Suppose a very small triangle ABC drawn on the surface of a sphere and all the points on its perimeter joined to O, the centre of the sphere. A small pyramid will be formed, of which the volume = ^(area of ABC) (radius of sphere). So, if any other surface be marked out on the sphere, it may be regarded as made up of a great number of very small triangles ; and if the points on its boundary be joined to the centre of the sphere, the volume of the solid thus formed = J area of the portion CALCULATION OF VOLUMES 451 of the spherical surface x radius of the sphere. In particular, the volume of the whole sphere = J (surface of the sphere) (radius) = J- . 4irr 3 . EXAMPLE i. What is the volume of the substance of a spherical shell, external diameter 12 in., internal diameter 8 in.? The volume = the difference between the volumes of two spheres of diameters of 1 2 in. and 8 in. respectively. .'. the volume in cubic inches = -7rxi2 2 x---7rx8 2 x- = ^{i2 3 -8 3 } 3 23 2 6 1 = 637 cub. in. approx. EXAMPLE 2. What is the volume of the spherical sector obtained by joining the ce?itre of a sphere 15 cm. in radius, to the points on the circumference of a small circle 0/12 cm. radius drawn on its surface ? Clearly, the distance between the centres of the sphere and the small circle = >Ji f - 1 2 2 cm. = 9 cm. Whence the spherical surface of the sector = 27r x 15(15 - 9) = iSoTr sq. cm. (p. 442), and the total volume of the sector = 1 x 15 x i8o7r c.c. = 9OO7T C.C. = 2827-4 c.c. EXAMPLE 3. A sphere, radius 9 cm., is divided into two portions by a plane, the distance of which from the centre of the sphere is 6 cm. What are the volumes of the two portions or segments ? In Fig. 96 let ABC represent the section, O the centre of the sphere. 452 A MODERN ARITHMETIC The volume of the spherical sector OABCD i.e. Q- radius x area of ABCD). The volume of the cone OABC = (J altitude x area of base) = lx6x77( 9 2 -6 2 ) c.c. The volume of the spherical segment ABCD = i x 9 x 2779 x 3 - J- x 6 x 77 (9 2 - 6 2 ) c.c. = 77(162 - 90) c.c. = 7277 c.c. EXAMPLE 4. Find the volume of the portion of a sphere, radius 12 in., comprised between two planes, distant 8 in. and 6 in. from the centre (and on the same side of the centre]. FIG. 96. FIG. 97. Here the required volume is the difference between the volumes of two spherical segments, i.e. between DA'B' and DAB. Volume of DA'B' .'. = ^x277xi2x6xi2 --i-77 Volume of DAB 2 - 6 2 ) 2 6 cub. in. X4xi2 -^(-v/i^^PfS cub. in. the volume of the required spherical frustum in cub. in. = 57677- 21677 - 38477 + - 77 _ 56877 3 CALCULATION OF VOLUMES 453 EXERCISES XCII. 1. Find the volumes, altitudes, or bases of the following cones : Volume. Diameter of base. Altitude. w ? 5-3 ft- 6ft. () ? 10-2 ft. 5ft- w 265-905 cub. in. P 12 in. (O 220-898 cub. in. ? 15 in. What is the content, in gallons, of each of the following vessels ? 2. A basin in the form of a frustum of a cone. Upper rim 24 in. diameter, lower rim 18 in. diameter, depth 6 in. 3. A tin can, lower portion cylindrical, 9 in. diameter, 14 in. high ; higher 5 in. 4 in. connecting portion a frustum of a cone 5 in. in height. 4. A vat 2 ft. deep, the top of which forms a rectangle 8 ft. x 6 ft., its base a rectangle 5 ft. 4 in. by 4 ft. 5. Show that if the earth is regarded as a sphere of 4000 miles radius, its volume is 268 x io 9 cubic miles approx. 6. Twenty small shot are dropped into a burette containing water, and they produce a rise of level in the water corresponding to an increase of volume of 0-281 c.c. : find the average diameter of the shot. 7. Find the diameters, volumes, or surfaces of the following spheres : Diameter. Volume. Surface. (a) 54 cm. p p <0 2-3 ft. p p (f) p 137-258 cub. in. p <*) ? 448-921 cub. in. ? (0 ? p 167-415 sq. cm. 454 A MODERN ARITHMETIC 8. A sphere, of 5 in. radius, is cut into two parts by means of a plane, distant 4 in. from the centre. Find the volume of the smaller of the two portions. (Take TT as -^.) 9. A spherical sector is obtained from a sphere of 6 in. radius, and the spherical cap bounding it has a height of 2 in. ; what is the volume of the sector ? 10. The interior of a glass goblet has the form of a portion ot a sphere ; the upper rim of the glass is 6 in. diameter, the base is 3-6 in. diameter ; what volume of water could the glass hold, the height of the glass being 2-4 in. ? 11. A spherical shell, external and internal radii 8 in. and 6 in. respectively, is cut in two by a plane bisecting at right angles a radius of the external sphere. What is the volume of the smaller of the two portions ? 12. A cylindrical boiler has hemispherical ends : its total length is 1 6 ft, diameter 6 ft. (inside measurements). What is its cubical content ? 13. A double convex lens has two spherical surfaces, both of 50 cm. radius, and its central thickness is i cm. What is its weight, the density being 2-3 ? 14. Find the cubical content of a mug in the form of a frustum of a cone, the upper radius being 2 in., the lower 2-5 in. and the height 5 in. 15. What is the mass of a conical bearing of iron, 6 in. long, diameter of the end 4-8 in.? (i cub. ft. of the iron weighs 450 Ib.) 16. A hay rick forms a frustum of a cone surmounted by another cone. The height of the frustum is 10 ft, the lower radius 10 ft., the upper 12-5 ft., while the height of the cone is 6 ft. What is the volume of the rick ? Graphic Calculation of Volumes. Suppose a solid to be divided up (Fig. 98) into several parts by a number of equidistant planes ; thus forming a number of thin slices, a typical example of which is seen in the middle CALCULATION OF VOLUMES 455 of Fig. 98. The slice being very thin its volume is obtained by multiplying its mean end area by the thickness, i.e. (A 4- A'\ j (thickness), approx. where A, A' are the areas of the two plane ends. FIG. 98. Draw an auxiliary graph (Fig. 98), such that the height of any plane section of (a) is marked along the line Oy, and the area of that section represented by a line perpendicular to Oy ; it is clear that, corresponding to the thin slice shown in (b\ we have a small trapezium (shown shaded in Fig. 98), the parallel sides of which are A, A', and the width is equal to the thickness of the slice. The area of this strip = (thickness) ( -Y Therefore the area of the strip represents the volume of the slice, and the area of the whole figure represents the volume of the whole solid (a). EXAMPLE i. Find graphically the volume of a cone, base i-inch radius, altitude 2 inches. Suppose ABC (Fig. 99) represents the section of the cone and" AO the axis. At a point N on the axis, the section perpendicular to the axis is a circle, NP being the radius. The area of the section is :rNP 2 ; but since TT is a constant multiplier it is convenient to disregard it for the present. 45 6 A MODERN ARITHMETIC At the corresponding point n of the axis Oy of the graph, set off a horizontal line to represent NP 2 on any convenient scale, and so for the other points. C O 1 FIG. 99. The area of the figure, multiplied by TT, gives the volume of the cone. Hence the volume = TT (0-66) cub. in. (By counting the squares.) EXAMPLE 2. Find the volume of a hemisphere of 2-inch radius. Suppose the hemisphere placed with its plane base hori- zontal ; the plan is a circle of 2-in. radius ; the contours are also circles. (Fig. 100, which is drawn to \ scale.) Set off the auxiliary figure, and mark the square of the radius of any contour against the height of the contour ; the constant factor TT need not be taken into account until the end. Count the squares. The volume= "y37T. FIG. ioo. EXAMPLE 3. Find the volume of a hillock from its given contoured plan, (i square represents io 2 sq. yds.} The contours (Fig. 101) give the areas of sections at heights of o, TO ft., 20 ft., 30 ft. above the plane of reference. CALCULATION OF VOLUMES 457 The areas of these sections are seen on counting the squares to be 399 x io 2 , 178 x io 2 , 61 x io 2 , 7 x io 2 sq. yd. ,0 1X1O* 2X1O* 3X1O 4 Areas in square yards FtG. IOI. Plot a graph of these areas against the heights; in Fig. 101 each square will correspond to the volume of a block of earth i ft. in height and io 3 sq. yds. in section, and, there- fore, to obtain the volume of the hill, count the number of squares in the figure and multiply by io 3 X9 to express in cubic feet. On counting, the squares are seen to be 431 in number. The volume of the hill is therefore io 3 x 9 x 431 cub. ft. = 3-9 x io 6 cub. ft. about. 458 A MODERN ARITHMETIC EXERCISES XCIII. Graphic. 1. Find the cubic contents of the vessels represented in section by (a), (b\ (c) in Figs. 102, 103, 104. (Scale i square represents i sq. cm.) FIG. 102. FIG. 103. FIG. 104. 2. Find the volume of the cask the dimensions of which are shown in Fig. 105. (Scale i square represents i sq. in.) (Scale i square represents i sq. yd.) FlG - r 5- FIG. 106. 3. Find the volume of the mound shown in Fig. 106, by its contoured plan. Exercises 10, 11, 13, 16 on p. 454 lend themselves to this graphic method of calculation. REVISION EXERCISES 459 REVISION EXERCISES XCIV. Miscellaneous: mainly from Examination Papers. A 1. A boy multiplied a number by 495 and obtained the product 1925617. He was told that the figures 9 and 7 were wrong, and the other figures right. What was the correct product ? 2. If a rectangle 43-86 yards long and 49-02 yards wide is divided into 129200 equal squares, how long (in inches and decimal of an inch) will the side of each square be? 3. A had a number of florins and B had the same number of half-crowns, and neither had any other money. B paid 3^. 6d. to A by giving 3 half-crowns to A and receiving 2 florins from A. If after the payment was made A had ^4. $s. 6</., how much money had B then ? 4. A person buys a triangular plot of ground, of which the sides are, respectively, 140, 143, 157 yards. Find its area in acres and the cost at 7s. 6d. per square yard. 5. If 500,000 rabbits bought in Belgium, at the rate of 75 francs per 100 are sold in England at an average price of lod. each, how much do the importers gain after paying 2 per cent, on their outlay for freight, etc. ? [25 francs = ,1.] 6. One man with his team could plough a field in 3 days. Another with his could do the work in 2\ days. In what time would the two men and the two teams together accomplish the work ? 7. What difference in amount of simple interest shall I obtain by lending ,5250 for 2 years at 3^ or 4^ per cent, per annum ? 8. A cistern 31 metres long and 4 metres wide holds water weighing 185752 kilograms. Remembering that I cubic decimetre of water weighs I kilogram, find the depth of the cistern in metres, decimetres, etc. 9. The yearly birth-rate of a certain town exceeds the yearly death-rate by three per thousand ; what time will elapse before the population increases by 50 per cent, (neglecting changes due to immigration and emigration) ? 460 A MODERN ARITHMETIC 10. The field CDFE from which the following measurements (given in links) were taken is bounded by straight lines. Draw it on the scale of I chain to the inch and calculate its area. B 220 114 F D 78 160 C 93 75 o 56 E A B 1. If the price of coal be raised 10 per cent, find by how much per cent, a man must reduce his consumption of that article so as not to increase his expenditure. 2. A capitalist, having a sum of money to invest, places half in a bank which offers depositors 4^ per cent, compound interest added yearly to the deposits ; and the other half in a bank, which also allows compound interest, where the rate is ^ per cent, less, but the additions of interest are made every six months. At the end of two years he finds there is a difference of ^249. i6.r. $\d. in the two accounts. Find the sum originally deposited. 3. A tradesman sells cloth at los. a yard, making a profit of 8 per cent, on the cost price ; if he sold the same cloth at ioj. 6d. per yard, what per cent, profit would he then make ? 4. A hollow closed cubical box is made of iron and is 2 inches thick, thfe external edge being 4 feet long ; it being given that iron weighs 7-1 12 times as much as water, and that a cubic foot of water weighs 1000 ounces, find in tons the weight of the box. 5. A merchant sold 90 cwt. 3 qr. 14 Ib. of madder, which cost him 38^. 6d. per cwt., to another merchant at the rate of 42^. per cwt., the purchase price to be one quarter in cash and the remaining three quarters in hemp: if the hemp cost the other merchant 24^. lod. per cwt., find how much money he would pay and the price at which he would reckon the hemp, on the supposition that he is to make the same net profit as the first, so that the exchange is equal. REVISION EXERCISES 461 6. Find, in cubic feet, the volume of the frustum of a cone, the height of which is 30 feet, the greater diameter 6 feet, and the less 3 feet (TT = Y). 7. A train, going at the rate of 72 miles an hour, overtakes another train 192 yards long, going in the same direction on a parallel line at the rate of 54 miles an hour, and completely passes it in three-fourths of a minute : find the time in which the trains would have completely passed one another, if they had been going in opposite directions ; find also the length of the faster train. 8. Find, with as little work as possible, the value, to the nearest shilling, of ,2-684 x (1-03 5 ) 4 . 9. From a cylinder of wood, with a diameter of 1 1 inches, a rectangular beam is cut. If the breadth of the beam is 8 inches, find the depth from the formula <tf 2 = D 2 - 2 , where b is the breadth and d the depth of the beam, and D the diameter of the cylinder. If the strength of a beam varies as &/ 2 , compare the strengths of beams 3, 4, 5, 6, 7, 8 in. broad cut from a cylinder 11 inches in diameter, and so find, roughly, the breadth of the strongest beam which can be cut from the cylinder. 10. A certain sum of money (say ^100) is put out at 10 per cent, compound interest, payable yearly. Draw a graph showing how the interest depends upon the time. Use your graph to find in what time the interest would amount to ,30. C 1. How many pieces of wire, each 4-13 in. long, can be cut from a rod whose length is 8-24 ft, and what will be the length of the piece left over? 2. Having given that a metre is 39-3708 in., show that the difference between 35 yards and 32 metres is less than a centi- metre. 3. One strip of carpet is i ft. \Q\ in. wide, and another strip is 2 ft. 3 in. wide. A square yard of the former costs M. more than a square yard of the latter, and a yard of the latter costs $s. 6d. How much does a yard of the former cost ? I 462 A MODERN ARITHMETIC 4. If ^ioo. 75. 6d. was allowed by a banker as simple interest on ,2555, deposited with him for ij years, what average rate pe cent, per annum of interest was obtained by the investor? 5. A can do a piece of work in 2^ days which B can do in 3^ days. If A's wages are i. i6s. 8^. a week and B's wages are ;i. Ss. 9</., what would A have charged for doing a piece of work for which B received ^n. 10.5-.? 6. A workman is able to save \i\ per cent, of his wages, but if his wages were raised 2s. a week and his expenses were increased by 10 per cent., his annual savings would be diminished by ijs. ^d. ; what are the man's weekly wages, a year being taken as 52 weeks exactly ? 7. A person subscribes the sum of 20 yearly to an insurance society, beginning on his 3Oth birthday ; to what sum would ne be entitled just before he attains his 55th birthday, reckoning com pound interest at 4 per cent, per annum ? log 1-04=0-0170333, ^26658 = 4-4258276, log 26659 = 4-42 5 8439. 8. Water has to be diverted from a river through a six-inch diameter pipe, running full bore at a velocity of one foot per second, to irrigate a field of 20 acres. How long will it take to deliver an inch of water over the whole area ? (Area of circle = 7rr 2 .) 9. ABCDEF is a field bounded by the straight lines AB, BC, CD, DE, EF, FA. The points B and C are on one side of AD, and the points E and F are on the other side of AD. A person wishing to find the area of the field, notes the position of points/, b, e,con AD, such that the corrssponding directions /F, B, eE, cQ> are all of them perpendicular to AD ; he also measures the lengths of these lines/F, B, etc., so obtaining the following data : A/= 8 chains, /F = 9 chains, kb = 10 chains, 6 = 7 chains, ke =23 chains, *E = 3 chains, be =25 chains, <:C = 8 chains. A = 29 chains. Draw a plan of the field and find the area in acres, etc. REVISION EXERCISES 463 10. The Great Wall of China is said to be 2400 km. long, 61 dcm. high, 457-5 cm. thick at the top, and 7625 mm. thick at the bottom. How many cubic metres of building material does it contain ? D 1. Divide 73438-6445 by 4-12345, and 0-004321 by 0-00043212, correct to three places of decimals. Add together 0-4 of a mile, 0-424 of a rod, 0-4246 of a yard ; giving the result (i) as a decimal of a mile, and (ii) in feet and the decimal of a foot. 2. In calculating the amount of wages lost by workmen in a strike, it was assumed that 200,000 men lost 42 days' work valued at 4.9. 6d. a day. If the error in the estimate of the number of men may be 10 per cent, of the true value, and that in the estimate of the value 20 per cent, of the true value, find the greatest possible error in the whole amount as a percentage of the true amount. 3. If ^12. 175. 3^. is divided among 4 men, 6 women and 8 children, so that each man may have twice as much as a woman, and each woman three times as much as a child, what sum will each man receive ? 4. If a piece of land in France cost 7350 francs per hectare, find its price per acre in English money ; 5 francs being equal to 4 shillings and 100 hectares to 245 acres. 5. A can just give B a start of 20 yards and C a start of 27 yards in a walking contest of a quarter of a mile. How much can B give C in 120 yards ? 6. A shopman bought eggs at the rate of 7 for a shilling, and sold them at a profit of 40 per cent. How many eggs would a customer get for a shilling ? 7. A closed cubical copper box, the sides of which are every- where three inches thick, encloses a cubical space whose edge is eight inches long. What would be the total cost of the material of the box at fourpence per cubic inch ? 464 A MODERN ARITHMETIC 8. If numbers are formed by arranging all the digits I, 2, 3, 5, 6, 7, 8 so that the odd and even digits occur alternately, show that each number is divisible by 1 1. 9. A sum of money is accumulating at compound interest at a certain rate per cent. If simple interest instead of compound were reckoned, the interest for the first two years would be diminished by 13^. 4^/., and that for the first three years by 2. os. qd. ; what is the 10. Find, in acres and deci- mals of an acre, the area of a field which is represented, on a scale of 5 feet to the mile, by ABCDEA (Fig. 107) ; or, if you prefer, take the scale as i (The figure is shown here at half size, i.e. 2*5 ft. to the mile.) FIG. 107. to looo, and give the area in hectares. If you want to draw perpendiculars, judge their positions by the eye. E 1. Find the square of i^f, correct to six places of decimals. Express the result as a decimal of 3-141593. 2. Find the value of 71 tons 7 cwt. 2 qr. at 30 guineas a ton. 3. At 6 a.m. on November nth, a watch which keeps fairly accurate time indicated 5 hr. 59 min., and at 6 p.m. on November 25th, it indicated 6 hr. o min. 56 sec. What was its daily gain ? 4. Two men undertake to do a piece of work for 6. One ol them could do it by himself in 16 days, while the other woulc require 20 days. They employ an assistant ; and, all three working together, they complete the work in 8 days. How much of the money has each of them earned ? REVISION EXERCISES 465 5. A train 99 yards long, and travelling at unitorm speed, overtakes in succession two trolleys on the other line of rails, one at rest and the other travelling in the same direction as the train. If the train pass the first trolley in 7^ seconds, and the second one 10 minutes later in 6| seconds, how far ahead will the train be when one trolley overtakes the other ? (The length of the trolley is to be neglected.) 6. Give careful instructions how to set out the boundary lines of a lawn tennis court 78 feet by 27 feet. 7. A cask is 30^ inches long, its bung diameter is 26^ inches and its head diameter is 23 inches ; find how many gallons of wine it contains. (Assume the form to be a double frustum of a cone.) 8. A man buys 500 eggs for i. 15^. and sells 200 of them at 8 a shilling. At what rate must he sell the remainder if he is to make 25 per cent, profit on his outlay? 9. Find, to the nearest penny, the compound interest on ^1257. i6s. in 4 years at i\ per cent, per annum. 10. The safe weight, in tons, which may be attached to a hempen rope of diameter d inches is given by o-gSjd 2 . Find the safe weights which may be attached to ropes of 0-5, i, 1-5, 2, 2-5, 3, 3-5 4 inches in diameter, and plot a curve on squared paper to show the nature of your result. Determine, by inspecting the curve, whether the following arrangements are safe : diameter 1-7 in., weight 3-5 tons ; diameter 2-8 inches, weight 7 tons. .PI. If 12 tons 2 cwt. 2 qr. 10 Ib. of coal is to be divided among 25 men and 20 women, giving each man three times as much as a woman, how much will each man receive ? 2. Find three numbers lying between 500,000 and 510,000, each of which is exactly divisible both by 37 and by 73. 3. On a map drawn to the scale of an inch to a mile, a certain estate covers an area of 2-44 square inches. Find the real area of the estate in acres. J.M.A. 2G 466 A MODERN ARITHMETIC 4. A landlord spends of the gross rent of a house in repairs and pays income-tax at lid. in the on the balance. If he then have ,85. ijs. 6d. left, what was the gross rent? 5. The interval between the ticks of a pendulum / cm. long is either ir\l seconds or 27rA/- seconds, where ^=981. Determine V V which ; given that, roughly, a pendulum a metre long beats seconds. 6. The New York Department of Agriculture has estimated the total farm wealth for the current year at $6,415,000,000. This total is stated to be 4 % higher than in 1904, 8 % higher than in 1902 and 36% higher than in 1899, each of these percentages being reckoned on the earlier year concerned. Calculate the total farm wealth for each of those years. Assuming that the percentage increase on 1904, as stated above, is correct to the nearest integer only, what are the limits of error in your result for that year ? 7. The distance from X to Y by a footpath is to the distance from X to Y by road as 5 to 6, and driving from X to Y by the road at the rate of 10 miles per hour takes i hr. 9 min. less time than walking from X to Y by the footpath at the rate of 4^ miles per hour. How far is it from X to Y by the footpath ? 8. The value of a piece of machinery depreciated yearly in such a way that the value at the end of any year was only 90 % of the value at the beginning of that year. The cost price was ^120, and it was sold eventually as waste metal for j. los. Obtain the number of years during which the machine was in use. 9. A cubical box, of external dimensions 17 inches each way, would contain crushed ore of the value of .421. 17.$-. 6d. if it were made of material I inch thick ; but by mistake it has been made of thicker material, and the difference in the value of the ore which it will hold is consequently .78. 17.?. 6d. ; what is the real thickness of the material ? REVISION EXERCISES 467 10. The diagonal AC of a quadrilateral garden ABCD is 975 links ; the perpendiculars on AC, from B and D, are respectively 385 and 567 links : find the area in acres, roods, perches. G 1. If a whole number be added to its square, show that their sum can only end in o, 2, or 6 ; and if a whole number be added to its cube, the sum can only end in o, 2, or 8. 2. The time of swing of a simple pendulum varies as the square root of the length of the pendulum. If a pendulum i metre in length swings once in a second, find to the nearest millimetre the length of a pendulum which swings 75 times in one minute. 3. What is the interest for two years to the nearest penny at 3! per cent, compound interest on ^2113. js. 7^.? 4. What is the value of 7 tons 15 cwt. i qr. 27 Ib. of copper at 58. 6s. M. a ton ? 5. State and explain the effect of moving the decimal point to the right or to the left of its original position. What is the value of 2-109375 of 2. 4J.? 6. A walks from X to Y at the rate of 4 miles per hour ; B, who starts 4^ hours later from X and rides on a bicycle at the rate of 12 miles an hour, reaches Y at the same time as A. What is the distance between X and Y ? 7. A cistern, whose length is 8 ft. 9 in. and breadth 7 ft. 4 in., is filled with water to a depth of 4 ft. 6 in. If a cub. ft. of water weigh 1000 oz., express as a decimal of a cwt. the weight of water in the cistern. 8. A pair or scales is improperly balanced, so that i Ib. in one pan balances 15-75 oz. in the other. If a merchant in selling goods places them in the latter pan, how much per cent, does he gain by such a proceeding ? 9. A man buys 76 gallons of wine at 5.9. fyd. per gallon ; he sells 13 gallons at a loss of 20 per cent., whilst two gallons are lost through leakage. At what price per gallon must he sell the remainder in order that he may gain 10 per cent, on the whole transaction ? 4 68 A MODERN ARITHMETIC 10. A person borrows ^1500, promising to repay the sum borrowed and the proper interest by ten equal yearly instalments, the first to fall due in one year's time. Reckoning compound interest at 5 per cent, per annum, find the value of the annual instalment. [(i-c>5) 10 may be taken as 1-629.] H 1. The metre was originally determined so that forty million metres should be equal to the earth's circumference : a metre is 39'37 inches. Compute from these data the mean radius of the earth in miles, being given that the circumference of a circle is to its diameter as 355 to 113. 2. A landlord's net rental, after paying 5 per cent, for collection, and 8</. in the pound income-tax on the remainder, is ^580 : find his gross rental. 3. Selling articles at 6s. 6d. each, a dealer makes a certain percentage of profit ; on increasing the price to js. id. his per- centage of profit is increased by 10. What profit is he making at the latter price ? 4. A triangular field ABC is to be levelled. The side BC is level and A is 2-3 metres higher than BC. Find how much the field is lowered at A, and how much it is raised along BC. 5. A litre of water weighs a kilogram, a litre of another liquid weighs 1-340 kilograms. A mixture of the two weighs 1-270 kilograms per litre. Determine the volume of each liquid in a litre of the mixture. 6. Employ logarithms to evaluate (0-41 568) 3 ' 25 . Find the side of a cubical block of iron weighing 7892 lb., assuming that a cubic foot of water weighs 62-5 lb., while iron is 7-726 times as heavy as water. 7. A sum of ^3826. loj 1 ., put out at compound interest payable yearly, amounted to ,5441. i$s. ^d. in 8 years. What was the rate per cent, per annum ? 8. A length of 60 centimetres is divided into equal parts. What is the number of these parts if, when this number is increased by unity, the length of each part is decreased by I millimetre ? REVISION EXERCISES 469 9. The Snowdon Tramway has a continuous rise of 3140 ft. in 4J{- miles. The average gradient is said to be i in 7-83. Compare these statements, and determine the true average gradient from the first statement to three places of decimals. 10. Draw, as accurately as you can, a fair sized plan of a field ABCDEF from the given Field-Book entries. Then find the area in acres, roods, poles. | Links. to E 500 to F 1700 3350 2760 2400 2080 1800 D 1000 to C 400 to B From A go North. I 1. (a) Reduce | - to its simplest form. 2 \4 ~~ I )~S\7 lO/ (&) Multiply 5-61177 by 0-71279 correctly to five places of decimals, employing by preference a contracted method of multi- plication. 2. How many square feet of timber I inch thick are required to make a closed box with a base 4 feet 6 inches x 3 feet, internal measurement, which shall contain 400 gallons of water? (Take 277-3 cubic inches to the gallon.) 3. What interest does a purchaser pay, if he adopts credit terms in buying the following article : " Military case will be sent on receipt of 6d. deposit and the payment of 17 weekly instalments of 6d. each, making 8j. 6d. in all, or 7$. 6d. cash." 4. How many feet in height is an isosceles triangle of which the base is 506 inches, and its area equal to that of a scalene triangle the sides of which are 368, 330 and 314 inches? 5. Divide by 17 lineal feet (i) 2023 lineal feet, (2) 2023 square feet, (3) 2023 cubic feet. State carefully what the quotient is in each case, and give reasons for your answers. 470 A MODERN ARITHMETIC 6. Find the cubical content of a block 2 ft. 9 in. long, i ft. 8 in. wide and i ft. 4 in. deep. If the weight of this block is n cwt., find what would be the length of a bar of the same material, having a sectional area off of a square foot and weighing 18 cwt. 7. From the equation t=2irij(l-s-g) find / in terms of the other quantities, and calculate its value to three significant figures when 8. If 15 cwt. 2 qr. 7 Ib. cost ^32. 13^. 7^., what will be the cost of 2 tons 3 qr. 21 Ib. 8 oz. at of the former rate ? 9. A circular basin of large dimensions, filled with water, is surrounded by a gravel path of uniform width : find the area of the walk and the diameter of its outer circumference, from dimensions of the paths only. 10. What sum to the nearest shilling paid now, together with ^300, ^250 and ^200 paid respectively at the ends of i, 2 and 3 years from the present time, will be equivalent to three sums of ^350 due at the ends of i, 2 and 3 years from the present time, if money is worth 3 per cent, per annum and compound interest payable yearly is reckoned ? J 1. (i) Express 2 r 2 - as a decimal fraction of 3-14159. (ii) Find what decimal fraction of 7 cwt. 2 qr. 13^ Ib. is equal to 5 cwt. i qr. 1 1 Ib. 3 oz. 2. A tank contains 9100 kilo, of water, and another tank of the same horizontal section, but 103 centimetres deeper, has a capacity of 10,750 litres. Find the depth of the first tank in metres, and the area of its section in square metres, (i c.c. of water weighs i gram : a litre is 1000 c.c.) 3. A rectangular area, the length being five times the breadth, contains 70,000 acres. How many miles of fencing would be necessary to enclose it ? 4. A cylindrical pontoon, 3 feet in diameter, having conical ends, is so far sunk under water that there is a mean pressure of 2-45 Ib. upon each square inch of its surface : find the total REVISION EXERCISES 471 pressure (in tons), the extreme length of the pontoon from the extremities of the cones being 14 feet and the height of each cone being 2 feet. (7r = - 2 7 2 .) 5. The number of first-class passengers is two-fifths of the number of third-class passengers, but the fares are respectively &d. and 2^d. How many are there of each class if the total receipts be 6. Find the compound interest on ^800 for three years, the rate of interest being 5 per cent, per annum. 7. A manufacturer sells to a merchant, who in his turn sells to a retailer. If the retailer sell the goods to the consumer for ,865. 3-y. o^., and each trader makes ten per cent, profit, what is the prime cost of the goods ? 8. What is meant by an average ? The following are the figures of twelve divisions of the House of Commons on a certain evening : Government, 201, 219, 255, 170, 195, 207, 223, 244, 250, 250, 248, 236; Opposition, 182, 205, 247, 101, 124, 136, 202, 243, 242, 235, 236, 183. What is the average majority of the Government for that evening ? 9. A man puts by \d. on the ist January, and doubles the deposit every succeeding year. How much will he have saved at the end of 25 years ? 10. Draw a plan of the field, and find its area in acres, roods, perches, from the following Field-Book entries in links : to D o 1050 o 840 250 to E to C 1 80 750 to B 320 600 520 450 to F o ooo o From A go North. 472 A MODERN ARITHMETIC K 1. In 1901 the National Debt was ^673,608,200. How much would be required to pay interest on this for a year at 2^ per cent. ? 2. Manitoba has an area of 47,188,500 acres ; of this 1,965,200 acres were under wheat in 1901, producing 18,353,013 bushels of wheat. What percentage was under wheat, and how many bushels an acre did it produce? Give each result to the nearest whole number. 3. (a) A and B run a mile race, and A wins by 60 yards ; after- wards B and C run a mile race, and B wins by 60 yards. If A and C run a mile race, by how much would A win, it being assumed that each runs at the same rate as before ? (b) A tub is filled with water by two taps running together in one minute forty seconds, and it would be filled by one of the taps alone in three minutes ; how long would the other tap, running alone, take to fill the tub ? 4. A steamer A is in distress and stationary, and fires a gun which is heard on another steamer B coming direct towards her. A fires another gun ten minutes after the first, and this is heard on B 9 mins. 48 sees, after the first. At this moment B is 4 miles from A. How soon will the two steamers be alongside of each other ? Give the answer to the nearest second, supposing sound to travel at the rate of 1130 feet per second. 5. Yellow brass consists of 66 parts of copper to every 34 parts of zinc ; spelter of I part copper to I of zinc. What amount of spelter would you melt with 324 tons of yellow brass, so as to produce an alloy suitable for sheathing, the composition of which is 6 parts of copper to every 4 of zinc ? 6. A frustum of a rectangular pyramid has a base of 27 inches by 24, while the top is 18 inches by 16, and the height of the frustum is 32 inches. Find its volume. If a plane passing through the smaller edges of the top and base divide the frustum into two wedges, find their respective volumes. REVISION EXERCISES 473 7. The yearly output of a gold mine decreases every year by 13 per cent, of its amount during the previous year. Given that the first year's output is ,260,000, and that (o-87) 10 = 0-24842 approximately, find (a) the total output for the first ten years ; (t>) the total output for all time. 8. To make an article takes 20 hours of a workman's time paid at 7%d. per hour. The material, 10 % of which is wasted in the working, costs lod. a pound. The final weight of the article is 20 Ib. At what price must it be sold to gain 8 % profit on the total cost ? 9. A and B are partners in a trading venture, A contributing ,2000 and B ^3000. A, however, acts as manager, the under- standing being that of the profits A shall receive 25 per cent, for his services as manager, the remainder to be divided in the ratio of their contributions. B's profits are ,120 ; what do A's amount to ? 10. The following table gives approximations to the interest which would accrue to ,200 in the specified times, the turnover being immediate, and the nominal rate 5 per cent, per annum. Time in years I 2 3 4 5 6 Interest in o 10-3 21-0 32-4 44-3 56-8 69-9 Exhibit the relation between interest and time, by means of a graph. Employ your graph to find (i) in what time the interest would amount to ^50-1 ; (ii) the interest which would accumulate in 3^ years 3 months. L 1. The product of two numbers, whose difference is 3, is 142504. Find the square root of 142504, correct to whole numbers, and hence, by trial, find the two numbers. 2. A rectangular vessel, 3*4 metres long and 2-6 metres wide, contained wine 7 decimetres deep. If as many casks as possible, each capable of holding 3 hectolitres, were filled from the vessel, 47 4 A MODERN ARITHMETIC how many such casks were there ; and what, to the nearest millimetre, was the depth of the wine left in the vessel ? (A litre = a cubic decimetre, and a hectolitre is 100 litres.) 3. A builder has two houses which cost him equal amounts ; he sells the first to A at a profit of 30 per cent., and the second to B at a loss of 20 per cent. A then buys B's house from him, giving the same price as he did for the first house. What profit per cent. did B make ? 4. The capacity of an oil can may be calculated approximately by regarding the lower part as a cylinder, and the upper part as a cone. If the total height is 14^ inches, the height of the cylindrical portion 10 inches and circumference 27 inches, how many gallons will it hold ? If the circumference of the can were a inches, what would be the circumference of another of the same shape but of double the capacity? (7r = 2 ^, I gal. = 277 cub. in.) 5. An express train 332 feet long and going at 62 miles per hour overtakes a goods train 526 feet long running in the same direction on a parallel line at 17 miles per hour. Find how many seconds the trains take to pass each other. 6. There is a row of 21 trees planted at intervals of one year. The youngest tree is 10 feet high and the middle tree is 25-9 feet high. Assuming that during each year of growth a tree adds to its height the same fraction of its height at the beginning of that year, find the heights of the oldest tree and of the third tree. (N.B. M 10 = 2-59 approximately.) 7. What would be the cost of the material of a closed cubical box, 3 inches in thickness on all sides, whose interior is also cubical and contains 27 cubic feet, reckoned at 2s. 6d. per cubic foot of material ? 8. What sum at compound interest will amount to ^650 at the end of the first year and 676 at the end of the second year ? 9. A dealer buys oranges of two qualities, one at a shilling a zen and the other at 8 pence a dozen. He then mixes them and lot at 15 oranges for a shilling. He makes a profit of 5 per cent, on his outlay : find the ratio of the number of oranges of the two kinds. REVISION EXERCISES 475 10. A rifle, sighted to 1000 yd., rests on a support 5 ft. from the ground and is fired. The height of the bullet above the support is given by the following table : Distance in yd. from firing point : TOO 200 300 4 00 500 600 700 800 900 IOOO Vertical height above support in ft. : 7-3 1 1 -2 I5-0 18-5 2I-O 23-3 25 22-5 16-5 On squared paper give a representation of the path of the bullet, showing also the ground level and the height of the support, and find in what positions a butt of height 20 ft. would stop the bullet. Ml. Find the number of gallons in a cylindrical vessel the diameter of which is 6 feet and height 7 feet. Given that the area of a circle equals Y times the square of the radius, and that a gallon is equal to 277-2 cubic inches. 2. A hare starts to run, at 12 miles per hour, when a dog is 44 yards off. After half a minute the dog sees her, and pursues at 1 6 miles per hour. How soon will he catch her? 3. Find the present value of an annuity of ^300 payable every year for 3 years at 5 per cent, per annum simple interest. 4. If the wages of 45 women amount to ^207 in 48 days, how many men must work 16 days to receive ^76. 13^. 4^., the daily wages of a man being double those of a woman ? 5. If the annual increase in the population of a state is 25 per thousand, and the present number of inhabitants is 2,624,000, what will the population be in 3 years' time ? And what was it a year ago ? 6. A man having 3 sons, left ^9656 to be divided among them in proportion to their ages at the time of his death ; when he died, their ages were 25, 22 and 21 years respectively; what was the share of each ? And what difference would it have made to each of the sons if the father had lived a year longer ? 476 A MODERN ARITHMETIC im r\f 7. A merchant commenced business with a certain sum of money, and at the end of each year increased his capital by 10 per cent. At the beginning of the fifth year 5 per cent, of his capital was worth ,732. is. What sum did he commence with ? 8. In 1891 the population of the East Riding of Yorkshire was 400,085, the Poor Rate ,166,260, and the number of paupers 11,133 ; in the North Riding the figures were, respectively, 354,382, ^167,574 and 10,483; and in the West Riding 2,464,415, .1,031,537 and 45,806. Find, to the nearest penny, the average cost of a pauper for the whole of Yorks ; and, to two decimal places, the average number of paupers per cent, of population. 9. ABCD is a quadrilateral plot of ground ; AB is 204 yards, BC 204, CD 253, DA 325, and the angle ABC is 60. Join AC : show that the triangle ABC is equilateral ; then find the area of each triangle in square yards. Give the area of the whole in acres. 10. A, B, C, D are four stations on a railway ; the distance AB is 10 miles ; BC, 10 ; CD, 8. The following is an extract from a time table : UP. A dep. 7.57 a.m. f arr. 8.15 a.m. I dep. 8.18 a.m. c /arr. 8.37 a.m. Idep. 8.40 a.m. D arr. 8.55 a.m. DOWN. D dep. 8.29 a.m. C B A arr. 9.10 a.m. Plot, in one diagram on squared paper, graphs to show the iitions of both trains at any time between those given, assuming that they run uniformly between the stations. When, and where, do they pass one another ? N 1. Two straight lines AB, CD, cut one another at right angles. If ACBD represent a four-sided field, find how many roods, poles. etc., there are in its area, the lengths AB, CD being 45 and 80 yards respectively. 2. Find the Least Common Multiple of 539, 1 183 and 1573. REVISION EXERCISES 477 3. How many doses, each weighing 0-00176 grain, can be obtained from a powder weighing 0-096 grain, and what is the weight of the remaining portion of the powder ? 4. Find, to the nearest integer, avoiding unnecessary figures in the calculations, the value of -' 7<396 x yo$m^ 0-7 x 0-0064 1 5. Two rectangular cisterns, the areas of whose bases are 36 sq. ft. and 45 sq. ft. respectively, stand at different levels, and are placed in communication by a pipe through which water flows from the first cistern to the second at the average rate of 3| gallons per minute. If a cubic foot of water weigh 62^ Ib. and 10 Ib. of water measure one gallon, in what time will the difference of the levels of the water be changed by 3 inches ? 6. With wheat at 28^. per quarter, 6 men can be fed for 1 5 days at a certain cost. How long can 7 men be fed for the same cost, when wheat is 30^. per quarter? 7. A racecourse is 1 1 miles in circumference, and 3 men start together to travel the same way round it. A travels 4 miles an hour, B 5^ miles an hour and C 8 miles. When will they all come together again ? 8- 33 per cent, of the candidates in an examination pass in mathematics and classics ; 42 per cent, fail in mathematics, and 52 per cent, fail in classics. What per cent, of the whole of the candidates fail in both mathematics and classics? 9. Find to within id. how much a trader should pay for 100 yards of silk so that, when selling it at 2s. n%d. a yard, he may make a profit of 35 per cent, on his outlay. 10. The three sides of a triangular field are, respectively, 85, 132 and 157 yards. Find its area in acres. It can be shown that this triangle is right-angled. Do this, and find the area again by an easier method. 47 8 A MODERN ARITHMETIC 66. EQUATION OF PAYMENTS. When various lots of goods are bought on the same date, but with varying terms of credit, it is frequently convenient to know the date on which one payment of the total nominal amount may be made, i.e. with no charge for extra credit and no deduction for discount. The question is equivalent to that of finding the average number of days of credit appertaining to each , and this number is known as the Equated term. EXAMPLE i. Find the equated term in the following : Goods to the value of 800, term of credit 62 days. 5oo, 6 1 The average number of days' credit each receives _ 800 x 62 + 600 x 31 + 600 x 31 + 500 x 6 1 800 + 600 + 600 + 500 _ 49600 + 1 8600 + 1 8600 + 30500 2500 _ 117300 2500 = 46-92 days. The equated term is therefore taken as 47 days, and the equated time as 47 days after the date of the transaction. More frequently the goods would be bought on different dates, as in the following example : EXAMPLE 2. Find the equated time in the following case: Goods. Term of Credit. April i $oo .2 months. April 10 600 3 5) J nl y ^ 500 2 It is desired to know the date on which a single payment of .1600 would clear the debt. EQUATION OF PAYMENTS 479 The sums are due as follows : June i, July 10, Sept. 15, and the example can be evaluated as in Ex. 1, working from some zero date (here conveniently June i). There are therefore bills of .500 due in o days, 600 39 from June i to July 10, ^"500 106 Sept. 15. The equated term _ 500 x o + 600 x 39 + 500 x 106 , 500 + 600 + 500 _ 23400 + 53000 _ 76400 1600 1600 = 48 days. The equated time is 48 days after June i, i.e. July 19 (use Table of Days). Instead of full payment of the sum total of the bills, the balance only may be required, certain cash payments having been made before the term of expiration of the various bills. EXAMPLE 3. find the equated time in the following example : Date. Goods. Term of Credit. March I 5 2 months April 1 6 600 3 June 7 S 2 April i, by cas July i, 750 Julys 400 2 Assume the same rate of interest throughout, remembering that cash payments tend to neutralize debts. Take March ist as zero date. Equated term _5oox6i+6oox 137 + 500 x 159 + 400 x 186-350x31 ~75ox 122 500 + 600 + 500 + 400 - 350 - 750 _ 30500 + 82200 + 79500 + 74400 - 10850 -9i5oo_ 164250 900 900 = 182-5 days, i.e. equated term = 183 days. Equated time is August 31. Balance ^900. 48o A MODERN ARITHMETIC EXERCISES XCV. 1. A person buys goods on April 10 to the amounts, and under terms of credit, as stated : Value of goods, ^550 ; credit, 2 mo. : goods, ,650 ; credit, 3 mo. goods, ^739 ; credit, 2 mo. : goods, ^640 ; credit, 2 mo. Find the average number of days' credit each receives, if the goods are paid for by a single payment of (^550+^650+^739 + ^640)- 2. A person buys goods as follows : Goods, 800. Credit, 2 mo. Goods, ^812. Credit, 80 days. ^736. 2 mo. .400- 3 mo - ^720. 70 days. Date of purchase June 12. Find the average number of days' credit each receives, if the goods are paid for by a single payment of the nominal sum of the values of the goods. 3. A person buys goods as follows : June i. Goods, 760. Credit, 2 mo. 20. ^650. 60 days. 28. .720. io.y. 80 days. July i. . ^364- -2 mo. Find the average number of days' credit from June i that each receives, as in Ex. 1 and 2. This average number is known as the equated term. The date obtained by adding the equated term to June i (or whatever the zero date be chosen) is known as the equated time. 4. In Ex. 3 find the equated term and time if the zero date be August i. EQUATION OF PAYMENTS 481 5. Find the equated time on account of the following purchases : (a) July 2. Goods, ^862. 5^. Credit, 80 days. ,760. 2 mo. .390. 2 mo. ^800. 60 days. 4- 15- Aug. 10. () May i. Goods, 61. 7s. 6d. Credit, 2 mo. 16. 76. 3-y. 6d. 60 days. June 3. ,59. i cw. 80 days. July 4- ^80. ioj. 2 mo. u. ^112.8^.6^. 2 mo. 6. Find the equated time of the following purchases and payments, and also the balance on account : (a} PURCHASE. PAYMENT. Date. Amount. Credit. Date. Amount. May 3 ?00 2 mo. June i ^1000 10 Soo 60 days June 14 7 $o 60 days July r ^1000 26 200 2 mo. July 3 600 2 mo. Sept. i ^500 PURCHASE. PAYMENT. Date. Amount. Credit. Date. Amount. Feb. i ^600 2 mo. June 3 ^I800 3 mo. Feb. 28 ^500 July 4 ^785 60 days June 30 ^1000 i? ^900 3 mo. July 31 ^1000 J.M.A. 2 H A MODERN ARITHMETIC to PURCHASE. PAYMENT. Date. Amount. Credit. Date. Amount. Jan. 10 ,500 2 mo. Mar. I ^600 Feb. 1 6 ^750 2 mo. May i ^800 Mar. 10 ;lOOO 4 mo. June 15 ^800 May 1 8 ^1350 3 mo. July i ^1000 PURCHASE. PAYMENT. Date. Amount. Credit. Date. Amount. Aug. 7 ^600 3 mo. Sept. 10 ^800 3 mo. Sept. i ^500 Oct. 12 ^900 2 mo. Oct. i ^500 Nov. 10 ^700 3 mo. Nov. i ^500 67. FOREIGN EXCHANGES. Money Exchange between different Countries. Different countries use different money systems; and accordingly it is important to be able to express any sum of money in the systems of different countries. For example, suppose a London merchant owes a Paris merchant 10,000 francs; he requires to know how many francs are equivalent to i. In calculating, it must be borne in mind that r st. The amount of gold in the standard gold coin of one country must always bear a definite ratio to the amount of gold in the standard gold coin of another country. But 2nd. The exchange value of the two coins fluctuates between certain limits. FOREIGN EXCHANGES 483 The tables below give particulars respecting the coinage of a few countries. COINAGE OF THE UNITED KINGDOM. The authorized coinage of the United Kingdom consists of the following pieces, exclusive of those issued only on special occasions : Denomination. Standard Weight. Least Current Weight. Remedy of Weight. GOLD : Grains. Grains. Grains. Sovereign - I23-27447 I22-500 0-20000 Half-Sovereign - 61-63723 6I-I25 O-I5OOO SILVER : Crown 436-36363 2-OOO Double Florin - 349-09090 1-678 Half-Crown 2l8-l8l8l * 1-264 Florin 174-54545 0-997 Shilling 87-27272 0-578 Sixpence - 43-63636 0-346 Threepence 2I-8l8l8 0-212 BRONZE : Penny I45-83333 2-91666 Halfpenny - 87-50000 1-75000 Farthing - 43-75000 0-87500 Standard Gold contains twenty-two twenty-fourths of fine metal and two twenty-fourths of alloy. The fractional amount of pure or fine metal in a ' gold ' or 'silver' coin determines its fineness. Since out of 1000 parts of standard gold, 916-67 are pure gold; or out of 24 parts of standard 2 2 parts are pure gold ; the fineness of standard gold is said to be 916-67 milliemes or 22 carats. A MODERN ARITHMETIC Standard Silver consists of thirty-seven-fortieths of fir metal and three-fortieths of alloy; fineness, 925. Bronze is an alloy of copper 95 parts, tin 4 parts and zinc i part. The Remedy is the amount of variation permitted in fineness and in weight of coins when first issued from the Mint. FOREIGN COINAGE. France. Germany. United States. GOLD: ioo franc piece double eagle 50 20 20 mark piece eagle (doppelkrone) 10 10 mark piece (krone) half eagle SILVER : 5 franc piece 5 mark piece dollar 2 2 half dollar > I quarter 50 centimes 50 pfennigs dime NICKEL : 20 pfennig piece 5 cents 10 5 BRONZE : 10 centime piece i cent 5 (ioocents = i dollar) 2 2 pfennig piece > ( ioo centimes = i fr.) 1 (ioo pfennig = i mark) FOREIGN EXCHANGES 485 The following are particulars respecting the standard coins in the above countries : British. French. American. Germany. Sovereign 20 Francs Eagle 20 Mark Weight in grains Troy 123-27 99-56 258 122-918 Fineness, i.e. proportion) of pure gold in total J 0-91667 0-900 0-8993 0-900 Weight of pure gold in j grains Troy / 113-0016 89-6 232 1 10-626 Florin 5 Franc Dollar Mark Weight of pure silver! in grains Troy f 161-45 347-23 37I-25 77-16 Fineness 0-920 0-900 0-900 0-900 Total weight in grains ) Troy / 175-5 385-81 412-5 85-73 EXERCISES XCVI. 1. Find the weight in grams of i sterling of the (a) standard weight, (b) least current weight. 2. Calculate the weight in grams of (a) is. and of (b) id. 3. What is the value per oz. Troy of (a) the gold used in coinage, (b) pure gold ? 4. What is the value of i oz. Avoirdupois of pure gold ? 5. How many sovereigns could be coined from 20 Ib. (Troy) of pure gold ? (State to the nearest sovereign.) 6. How many shillings could be coined from 2 Ib. (Troy) of pure silver? The following (Exx. 7, 8 and 9) are quotations from mining returns. Find the fineness of the gold from the mines mentioned : 7. ASHANTI GOLDFIELDS AUXILIARY. The clean up for April from Dredges Nos. 4 and 5 is as follows : Gold recovered, 194 oz. 4 86 A MODERN ARITHMETIC (approximate value ^776) ; total recovered for four months ending April 30, 1906, 885 02. (Approximate value ^354-) 8. N. BONSOR MINE. The tributors crushed 18,576 tons of ore at the Bonsor battery, yielding 10,525 oz., and 12,422 tons of tailings were treated, yielding 4413 oz. Troy, or a total of 14,938 oz., of a value of ,42,687. 9. BONSOR AMALGAMATION. The tributors ran the mill for 179 days, crushing 15,784 tons, yielding 3505 oz., and 8906 tons of tailings were treated, yielding 1451 oz., or a total of 4956 oz., of a value of ,14,060. 14^. 6d. 10. Find the value in English money of the gold in (a) a 10 mark piece, (b) a 10 franc piece, (c) an American eagle. 11. Calculate the fineness of the standard coins of the countries named in the following tables : (Great Britain sovereign, weight 123-27 grains, ff fine.) Country. Gold Coins. Legal Weight in Grains. Sterling Value. . s. d. (*) Argentine Argentine or 5 124-44 o 19 10 Republic peso piece 0*) Austria-Hungary Ducat 53-85 094 fc) Brazil - 10 milreis 138-35 I 2 5^ (</) Denmark - 10 krone piece 69-14 II 0| ( g ) Egypt loo piastre piece I3I-I8 i o 3^ (Egyptian ) (/) Japan - 20 yen piece 257-21 2 Ilf Or) Mexico 10 peso piece 26I-12 205! W Ottoman Empire Turkish pound of III-36 o 18 of (0 Persia loo piastres Toman of 200 57-90 o 9 5 shah is C/) Peru - Libra of 10 sols 123-27 I O O (*) Russia Imperial of 15 199-10 i ii 9 roubles FOREIGN EXCHANGES 487 12. What would be the value in francs of the gold in an English sovereign (weight 123-27 grains, 22 carat fine), the 20 franc piece being 99-56 grains in weight and 900 fine? In other words, find the Mint gold par of exchange between England and France. 13. Using the tables in Ex. 11, find the Mint gold par of exchange between Great Britain and (a) Russia, (b) Japan, (c) Denmark. 14. If only 25-11 francs be regarded as the Paris equivalent of ,1 sterling in London, it just pays to ship gold from London to Paris, i.e. to send across English sovereigns and have them re- coined into French money. If 25-33 francs in Paris are equivalent to ^i in London, it just pays to ship gold from Paris to London and have it recoined there. These values for the exchange are known as the Specie Points. The actual course of exchange between London and Paris generally varies within these limits. Find (a) the cost per sovereign of transmitting gold from Paris to London and coining, or of transmitting from London to Paris, also (b) what would be the cost of so transmitting ,35,700. 15. If the cost of transmitting gold to Berlin, and having it recoined there, be 11 ; when the amount transmitted is .2042, what are the specie points between London and Berlin ? 68. FOREIGN BILLS OF EXCHANGE. Even when the buying and selling of goods take place in one country, money payments are seldom made, except when in one town, and when the amounts are comparatively small; more usually cheques, bills, etc., are employed instead. In dealing with foreign countries, similar or modified methods are employed, except that cheques are most fre- quently not now available. Also, it is clear that since trading between two countries consists generally in an exchange of goods, it is not actually necessary that much money or bullion should pass at all between two countries trading with one another. 4 88 A MODERN ARITHMETIC Thus the table below gives for the United Kingdom the actual imports and exports of bullion and specie, also the value of imports : Imported Bullion. Exported Bullion. Imports. Year 45,261,468 35,491,184 413,434,242 I8 99 39.513,173 3 I 972>39 523,075,163 1900 32,217,306 26,015,102 521,090,198 1901 31,393.345 26,125,206 528,391,274 1902 28,967,723 39,233.238 542,600,289 1003 The following examples illustrate two of the most common methods which may be employed by a person in London or England in liquidating a debt abroad, say at Paris : EXAMPLE i. Calculate the cost of settling a debt of 12,000 francs by buying from a bill broker, 3 months' bills on Paris */ 25-50, discount 3!%. Debt= 12000 fr., if paid now. 3 months' interest at 3 J % = 105 12105 = the equivalent of debt in 3 months, at the end of which time the bills mature. In addition there must be paid Brokerage at i per mille* = 12-11= fees charged by the btamp duties \ per mille = 6-06 bill broker. 12123-17 fr. total. Amount to be paid to settle the debt 12123-17 =^475-416 2 5'5 = ^475- 8 -f- 4^- to the nearest penny. *i per mille is i per 1000 or o-r %. FOREIGN BILLS OF EXCHANGE 489 NOTE, i paid now buys bills to value 25-50 fr., or more than the par rate of exchange, on account of the fact that 25-50 fr. is the value of a bill maturing in 3 months' time, which can be bought for i now. The exchange, however, depends upon the relative demand of London on Paris, and vice versa, and not on the interest alone. EXAMPLE 2. Find the rate of cost df settling a debt by em- ploying an agent in Paris to draw a 3 months' bill on the debtor. The agent draws up a 3 months' bill or draft for the debtor to accept ; this bill the agent then negotiates in Paris. The rate of exchange will now be less than the par rate, say 25-17 fr. i.e. 25-17 fr. is now the present value in Paris of each i on the draft, payable in London in 3 months. By expending \ in 3 months' time, the debtor therefore can liquidate a present debt of 25-17 fr., less, however, by 0-02517 on account of brokerage and ,, 0-012585 stamp duties. .'. the net amount is 25-132245 francs. A Present Payment of i at the same rate of discount as in Ex. 1 would liquidate 25-132245 fr., increased by |-% (the percentage increase at 3 months at 3^ % per annum), i.e. i now would liquidate a debt of (25-132245 + 0-219907) fr. = 2 EXERCISES XCVII. 1. Find the cost to a London merchant of settling debts of (i) 385,000 fr., (ii) 43700 fr., (iii) 58200 fr., assuming the following rates of exchange for 3 months' bills : (a) London on Paris 25-47. (b} London on Paris 25-5. (c) Paris on London 25-12. (d) Paris on London 25-14. Rate of discount 3%. Brokerage i per mille. Stamp duty 0-5 per mille. 490 A MODERN ARITHMETIC 2. Find the cost to a London merchant of settling a debt of 450,000 marks, assuming that the rates of exchange for 3 months' bills are as follows : London on Berlin (a) 20-63, (b) 20-52. Berlin on London (c) 20-36, (<i) 20-33. Rate of discount, brokerage and stamp duties as in Ex.1. 3. Find the rate of discount when a 3 months' bill, ^1236. i is. y endorsed 25-18, is worth ,1227. or. \d. (Brokerage and stamp duty as above.) 4. Find the exchange in London on Paris for 3 months' bills, if a debt of 12617-54 fr. can be liquidated for ^500. (Rate of dis- count 2%, brokerage and stamp duty as above.) 69. STOCKS AND SHARES. In medieval and earlier times, financial companies of the type met with now could not exist, and for many reasons : (a) laws were in force against usury ; (b) no opportunity existed for storing up wealth, other than in the form of corn, cattle, and other perishable commodities ; (c) very little business was carried on between town and town ; (d) most of the labour was expended upon material furnished by the consumers, and therefore a large amount of capital was unnecessary. These conditions gradually changed however; the restric- tions with regard to lending money broke down ; stores of the precious metals were discovered, towns became less isolated, while capital became more and more necessary for carrying on wars and for other purposes. The goldsmiths started banking business, and banking firms gave interest to small depositors. The earliest borrowers were kings, the money being expended generally upon wars, and to a smaller extent on Public Works. In this way arose the National Debt. At the time of the STOCKS AND SHARES 491 Revolution in 1688, the National Debt was only ^666,400; a terminable annuity, charged upon certain branches of revenue ; increasing after the Ten Years' War, it amounted to ^14-5 millions in 1697 ; at the end of the reign of Queen Anne it stood at 36-1 millions, while at the present time it is about ^800 millions. The early rate of interest was high, but gradually it has become less, while attempts have been, and are being made to reduce the amount of the debt by sinking-funds and other means. Examples on page 504 furnish illustrations of this. In addition to the money invested in all the securities comprised in the National Debt, money was invested in various companies. The earliest companies were known as Regulated Companies, each member trading on his own capital, but observing certain rules (somewhat similar to the Stock Exchange at present). The members could compete against one another; but the membership offered much prestige, and many advantages with respect to tolls, customs, and recovery of debts. Later, many of the Regulated Companies became converted into Joint Stock Companies. A Joint Stock Company is a single corporation, with one capital, and trading as a single individual. The most famous of the early Joint Stock Companies was the East India Company, formed as a Regulation Company in 1600, and afterwards converted into a Joint Stock Company, as the distance to the East Indies made the risk of trading too great for a single individual. Other well known companies were started, but not with quite the same success. The " South Sea Bubble " was founded to trade with Spanish America, in the hope that large profits might be made from the slave trade and the whale fishery. In April 1720 the 100 shares of the Company stood at 120, in July at 1020, but 492 A MODERN ARITHMETIC eventually became practically worthless. The company being Joint Stock, each member was liable for the debt of the company; many thousands of people were ruined by the bursting of the " Bubble," and shortly afterwards it was made illegal to form companies with transferable shares, unless the company was incorporated by Royal Charter. In these new companies members were only liable for their contributions to the Share Capital. Later, companies of this kind were abolished, and Joint Stock Companies became again all prevalent; but in 1855 public opinion once more deemed Limited Liability reasonable, and now there exists a very large number of Limited Liability Companies, in which the individual member is not responsible for the debts of the company, his responsibility ceasing entirely when once his shares are fully paid up. With reference to various companies and stocks, the following terms may be noted in addition to "Joint Stock Company " and " Limited Liability Company " : Ordinary Stock. When the capital of a company, or amount of loan, is divisible for the purpose of sale into amounts denoted by their nominal money value, it is spoken of as Stock. Thus Consols are Stock. Sometimes any amount of Stock can be dealt in; e.g., you may buy or sell ^68. 13.?. 7^. nominal of Consols ; in other cases, dealing can occur only with multiples of i, $, and so on. Shares. When the capital is divided into a certain number of parts, these parts being indivisible, then these parts are known as Shares. Preference Stock is stock that has priority, or preference, over ordinary stock, not only with regard to dividend, but also on liquidation. Bonds are documents, in which the issuer promises to pay a sum of money and interest at some time, either fixed in STOCKS AND SHARES 493 advance, or by drawing by ballot the number of the bond (according to the terms on issue). Debentures are bonds, the repayment of which, together with interest, is secured by a charge upon the assets of the company which issues the Debentures. Scrip (short for Scrip Certificates) are provisional docu- ments which are issued by companies and others, as evidence that the holder is entitled to receive stock, bonds, etc., when issued. In 1694, the Stock Exchange came into existence, by means of which people might exchange their money for stock, or stock for money. In fact, the Stock Exchange was then, as now, a Market in which various shares or stock are bought or sold ; these shares or stock are virtually sleeping partnerships. To the Stock Exchange market ordinary people may not go : their transactions are effected by intermediaries or agents, known as Stock Brokers, and through them with buyers or sellers known as Jobbers. These dealers or jobbers are generally willing to sell or to buy, but, as in all other cases with people who buy or sell, the two prices are different ; i.e. the jobbers are willing to sell stock at one price, but only to buy at a slightly lower price. Examples ; furnished from extracts from a daily paper : (a) " Consols "( 2 i%) 9<4 9<>f. (b) "Brighton" 134 136. (*j " Aerated Bread " 6| 6|. (a) Means that the jobbers offered to sell to the brokers ;ioo Consols (note carefully ^100 Consols does not mean .100 money) for ^Qof sterling, or offered to buy each ;ioo Consols for ,90 \ sterling, and that the holder of each ;ioo Consols receives dividend at the rate of 2% per annum. 494 A MODERN ARITHMETIC () Means that jobbers offered to sell Brighton Railway Stock at ^136 sterling for every ^100 stock, or to buy Brighton Railway Stock and pay ^134 for every 100 Stock. (c) Means that for one share in the Aerated Bread Company a jobber asked ^6|, but that for one share he only offered to pay ;6f . EXERCISES XCVIII. a. STOCKS AND SHARES. BRITISH FUNDS. Consols 2\ per cent. - - - 90^ 90$ Ditto, May Account - - - 90^ 90}^ New Transvaal Loan - - - 99! 99? Local Loans 3 per cent. ... 996 99! India 3 per cent. - ... g6| 96! Ditto, 2 \ per cent. - - - 81^ 8if Ditto, 3| per cent. - - - 1051 105! Rupee Paper 3! per cent. - - 66 66| Irish Land Stock - ... g 2 | 92! National War Loan 2| per cent. - 98! 98! Water Board 3 per cent. Stock 95 1 95^ FOREIGN GOVERNMENT STOCK. Argentine, 1886 - IO 2i 103 Buenos Aires Waterworks 5 per cent. 102^ io2f Brazilian 4 percent, 1889 - - 87 87! Ditto, West of Minas - - - 99! 993 Chilian, 1896, 5 per cent. - - JO o| 101} Chinese 5 per cent., 1896 - - 103 103^ Ditto, 5 per cent. Railway Loan - 102^ 103 Egyptian Unified - - - - 106 106* French 5 per cent. Rentes - - 97 99 German 3 per cent. - - - 86 86J Greek 4 per cent. Monopoly - - 53! 54 Italian 5 per cent, taxed - 104! io4f Japanese 6 per cent., 1904 - - 101 101^ Ditto, 6 per cent. (Second Series) ioo| 101} STOCKS AND SHARES 495 HOME RAILWAYS. Brighton 134 136 Ditto, Deferred - - - - i2o| i2i Caledonian - - - - uoi m Ditto, Deferred - 36^ 37 Central London - - - 90 J 91^ Chatham and Dover - 15^ 15! Ditto, 2nd Preference - - 6oJ 6i Furness Ordinary 63 66 Great Central Preference - - 39} 39! CANADIAN AND FOREIGN RAILWAYS. Antofagasta and Bolivia - - 221 223 B.A. Great Southern, - - - 137 138 MISCELLANEOUS. Aerated Bread 6f 6f Allsopp New Ord. - - - - 20 21 Ditto, Pref. - 34 38 Anglo-American Tele. Ord. - - 66 68 Pref. 114 115 E>ef. - - 23! 24 Apollinaris 6| 7| Armstrong, Whitworth - - - 58/6 59/6 MINING MARKET. AFRICANS. Anglo-French - - i Boksburg - - ]% Bonanza - - - . City and Sub. - - 4.1 4! Kleinfontein - - i i$ o 4- Knights - - - 3! 3! Langlaagte - 2 f 2 f May Consolidated - 2\^ 2.\{ Modderfontein - - 7^ 7^ 496 A MODERN ARITHMETIC Use the foregoing list, and find : 1. What would be the price asked for by a jobber in British Funds for (a) ^1500 Indian 3 per cents. (0) ^2300 Irish Land Stock. (c) 1200 Water Board 3 % Stock. 2. What a jobber would offer to pay for the following Foreign Government Stock : () ^2350 Argentine, 1886. (b) 4100 Chilian, 1896, 5 %. (c) ^5000 French 5 % Rentes. 3. (a) If a jobber buys and sells 1760 shares of the Aerated Bread Co. as above, what profit he makes by the transaction. (b) What is the difference between the buying and selling price of (i) 2760 shares in Apollinaris ? (ii) 3725 Armstrong, Whit worth ? (iii) 160 mining shares in the Bonanza? Brokerage and other Charges. Since the dealings with a jobber are always conducted through an intermediary stock-broker, certain charges must be paid, whether for selling or for buying stock. These charges vary with the value of the stock or share. They are regulated partly by competition, partly by custom ; generally they are : J% in the case of Consols, i.e. \ for every 100 stock bought or sold ; J% in the case of railway stock ; 3^. per i share (nominal) in mining shares. In addition to these com- mission fees, further charges are incurred, as will be evident from the procedure described below. STOCKS AND SHARES 497 The stock is bought from the jobber by the broker, and On a certain day a Transfer Deed is prepared by the broker, Showing that the actual seller (the holder of the stock at present) is to transfer the stock to the buyer; and within 10 days after, the buyer has to pay, and the transfer deed is delivered. This, after being signed by the buyer, is sent back to the broker, who sends it on to the company whose stock or shares have been purchased ; they note the transfer and send the certificate back. For the transfer deed, a small amount is paid, and certain stamp duties, varying with the amount of the transfer and its value. In the following examples no account is taken of stamp or transfer duties : EXAMPLE i. Find the- cost of buying ^5700 stock n Chinese 5 %, 1896 ; 103-103 J. Brokerage J %. Here ,100 stock is boCight from the jobber for money; the broker's fee bring'S its cost up to .'. the cost of ^"5700 stock = ^'-~- x 103! money EXAMPLE 2. What amount of French $ % Rentes, 97-99, tan e purchased for ,3980? What yearly dividend would the .holder be entitled to, and what would be the rate of interest .received on his money 1 Brokerage \ % . Here .(99 +|) money would purchase ^100' Rentes/ yielding ^5 yearly, -" >S9 8 would purchase ^roox^- ' Rentes, yielding 39 8 99 ' 5 .*. the amount is ^40^0 Rentes and the income ^200 yearly. The rate of Interest is - 2 -%* i.e. 5-025 % approx, 39' 8 J.M.A. 2 1 498 A MODERN ARITHMETIC EXAMPLE 3. What change of income would be occasioned to an investor on selling ,10,000 Chilian 5 % stock^ ioo|-ioi|-, and investing the proceeds in Japanese 6%, loi-ioij? Brokerage J % on each transaction. On the sale of ^"10,000 Chilian 5 %, , ,. 10000 ;(ioo-i) x- money, i.e. ,10,025, would be received. The income on this if invested in Japanese 6 % ' ^ould be , . X(S P er annum ' ' The income originally was x 5, or ^500. The change = ^9 1. 35. id. EXERCISES XCVIII. b. Find the cost or proceeds of the following transactions (brokerage. | % in all cases) : 1. Buying ,1230 Consols a.'c 9i|-|. 2. Selling 580 L. & N. W.R. at 143^-4. 3. ,1000 G.E.R. at 104-105. 4. Buying ,20,000 G.W.R. at ii4|-|. 5. ,: 1 80 Consols at 91 1~|. 6. Selling ,380 L. & N.W.R. at 143-145. 7. Buying ,4200 G.E.R. at 87^-88. 8. Selling 6000 Argentine at 74|-f. 9. Buying ,980 Stock 46|-|. 10. Selling ,10,000 Stock at 98^-99. 11- 1120 at ii 2 i-|. 12. Buying ,6000 at 9 6i-|. 13. Find, correct to 3 places of d per annum on the money investe-' 'J^^ ^ nt t( * P * C ^ full dividend paid (brokerage ne A m ^' followm ' assummg th. glected: 111 all cases) : STOCKS AND SHARES COLONIAL GOVERNMENT SECURITIES. 499 Stocks. Closing Pi ices. (a) Canada 3 % Reg. 97i 98* <*) Cape of G.H. 3^ % Ins. - 97i 98i (% Natal 3 % Ins. - 86J 87i (<*) N.S.W. 3i%, 1924 - - 99i 100^ w 3% Ins. ' - 89 90 GO N. Zealand 4 % Ins. - 107 109 Or) 32 % Ins. 101 102 (A) 3 % Ins. - 89ir 9i (0 Queensland 3 % Ins. 88| 89^ 0) Rand Water Board 4 % - 95 97 <*> S. Australia 4%, 1916-36 - 101^ 102^ (/) Tasmania 3^ % Ins. - 100 101 o) Victoria 3^ % Ins. ioo| IOI^ () 3 % Ins. - 90 91 (*> W. Australia 3%, 1915-35- 8 7 | 88| 14. Find the. change in the income due to selling out from the stocks in the table in Ex. 13, and reinvesting as follows, assuming brokerage j % in each case : (a) ,12,000 Canadian 3 % (b) ,28,000 N. Zealand 4 % (c) ,56,000 Tasmanian (d) ,19,500 N.S.W.. 3% (*) .2500 Natal 3 % into N.S.W. 3| %. W. Australian 3 %. Victorian 3 %. Rand Water Board 4 %. Victoria 3%. (/) ^12,250 Cape of G.H. 3 |% Cf) ^ 2 35 Canadian 3 %. (//) ^36,000 Queensland 3 % (/) ^12,500 New Zealand 3 % (/) ^80,000 Rand Water Board Cape of G.H. 31 %. New Zealand 3 %. Queensland 3 %. New Zealand 4 %. State your answer to the nearest shilling. soo A MODERN ARITHMETIC 15 The yield per cent, on the money invested in the following Indian Railway Companies is stated below ; find the price to the nearest shilling of the stock : (*) (c) 00 to Railway. Dividend per cent. Yield per cent, on Money. Price of Stock. Assam Bengal - 3 ^. 3J. IQd. ? East Indian 4 2. I9J. 3d. ? Madras s 4. os. 6d. p Southern Mahrattas - s 4. i8j. a/. p Gt. Indian Peninsula 3 2. iss. *>d. p Bombay, Baroda s 3. 6s. od. p 16. What was the percentage declared on the following Railway Stock, if the rate of interest on the money invested was as in the accompanying table ? 00 (*) to ('0 to CO Or) Stock. Approx. Price. Dividend for 1905. Per cent. Yield per cent. Great Western - i35i p 3. 19*. od. Brighton Def. - 1 20 p 4. 7s. 6d. Lane, and York. 105 ? 3. IQS. od. Midland Def. - 67 p 3- 17J- d. North-Eastern - 140 P 3- I 7s- bd. North- Western - 156 ? 3. igs. od. South- Western Def. - 52* p 3- 15^-6^ 17. State equivalent percentage returns on the stocks (a) to (/) (all supposed of the same security) : and find the equivalent % on stock at the prices given on the right. 00 z|% ; price of stock no, <*) 5%; 155, to 3%; . 44, 00 9%; 198, to ?4% ; ioo, (/) ^3- I2j.cx/. % ; ioo, ..(i) 1 60, (ii) 220. ,..(i) ioo, (ii) 771, (iii) 248. ...(i) ioo, (ii) 8o, (iii) no. (i)66, (ii) 13?. ...(i)4o, (ii) 731, (iii) 93s- (i)75lj (ii) IIO J (111)165. STOCKS AND SHARES 501 18. A Forest, Land c. Co. advertises for subscriptions for 450,000 six % Preference Shares of i each, 400,000 five % First Mortgage Debentures of ,1, and a certain number of Ordinary Shares. If the gross profits be estimated at ,238,000, and if from this there be paid oui for general expenses, reserve fund, etc., the sum of ,158,000 (not including interest on Debentures), and if the interest on the Ordinary Shares be estimated at 1 1 % on the nominal capital, how many of such shares are offered for subscription ? 19. A Motor Car Company offers for subscription 100,000 seven % Preference Shares of 1 each ,100,000 100,000 Ordinary ,100,000 200,000 The Company, although it anticipates a profit of 3^. per mile run per day, is content to estimate a profit of 7.d. per mile per day, with runs of 100 miles per day, for each of 100 motors, and an average of 300 days to the year. It allows 3000 for administrative expenses. Calculate the net estimated profits ; also what approxi- mate dividend could be declared on the Ordinary Shares if 5000 be put by as a reserve fund. 20. On a Company being floated, a man receives an ' option ' for the 'call' of 3000 shares at if within 3 mo. When the shares stand at ij$-2, the man exercises the right of his option, and sells 3000 shares at the market price. The shares necessary for the sale are supplied by the Company in accordance with the terms of the option. What does the man make? (Brokerage 3^. per share for the sale.) N.B. The * call ' entitles the receiver of it to the right of calling upon the giver to sell to him, within or at an agreed time in the future, at an agreed price, a certain number of shares. 21. What does a man make by exercising his option for a call at 2| for 2500 shares, when the shares stand at 2-}f-2^f, if he has paid 200 for the option, and if brokerage be estimated at 3^. per share on the sale ? 502 A MODERN ARITHMETIC 22. A man pays ,120 for the option of a ' Put,' i.e. for the right to call upon the giver of the option to buy from the receiver, at an agreed time in the future, a definite number of shares, or a definite amount of stock, at a fixed price. The option is for 3600 shares at 1$ ; the option is exercised when the price of the shares in the open market is iiV-ifV- What is the gain or loss due to the exercising of the option (brokerage 3^. per share)? 23. " " Tramways and Carriage Company. For the year 1905 the gross receipts were ,256,741 and the expenses .171,661. A dividend at the rate of 9^ per cent, per annum is proposed for the last half-year, making, with the interim of 8^ per cent, per annum, 9 per cent, for the year. (An interim dividend is one declared in the middle of a working year.) The sum of .12,053 is added to the reserve fund. If the balance was used up in paying dividend, what was the share capital, and what was the sum paid as dividend the first half-year ? 24. Great Central and Midland Three and a Half Per Cent. Guaranteed Stock. Issue of ,2,000,000. Messrs. Glyn, Mills, Currie & Co., as bankers of the Joint Committee, are authorized to receive subscriptions for the Stock at the price of ,103 per centum. Payable on application - , . . - 5 per cent. On allotment (the premium of ,3 and an instalment of ,20 of capital) - ,. - , - - 23 1 5th May, 1906 - - 25 1 5th August, 1906 - . . , 35 1 5th November, 1906 - - , . .35 103 A man buys stock calculated to yield him ,250 per year ; what are his several instalments ? 25. " R. and J. H ." The accounts show a profit of ,19,348 on the year's trading, and an available balance of .15,683. The STOCKS AND SHARES 503 directors recommend that ,1250 be added to the reserve account, that a dividend be paid at the rate of 4 per cent, per annum, and that the balance, ,1356, be carried forward What was (a) the share capital, (b) the market price of a ,1 share ; if the interest earned on the actual money invested was 4\ % ? (The payment to the reserve account, together with the amount carried forward, is taken from the available balance.) 26. London Electric Supply. The quantity of electrical energy sold in 1905 amounted to 13,042,932 units, producing a net revenue f .78,721. From this a sum of ,30,000 was carried to reserve accounts, and 3141 carried forward. The directors proposed a dividend of 4 per cent. What was the (a) share capital, (b) the net profit per unit ? 27. Harrod's Stores. The directors recommend a dividend of 15 per cent, and a bonus of 2 per cent, for the six months ended January 31, which, with the interim dividend, will make 22 per cent, for the year, leaving ,22,261 to be carried forward. Find the net interest received for the year on the money invested by a buyer of the i share at the current price. Harrod's Stores 4^-4 1. (Brokerage, 6d. per share.) 28. A mine is estimated to give a net profit of ,2500 yearly for 20 years. The owners endeavour to float a company with a certain number of i shares. How many shares should be issued, on the understanding that the yearly profit will be enough to repay the money invested at the end of 20 years, with interest at the rate of 5 % per annum ? (N.B. A yearly payment of .8. os. $%d. for 20 years would repay ,100 with the interest thereon at 5 % per annum.) 29. Home and Colonial Stores. The net profits for the year amounted to .133,419, which with the balance brought forward makes ; 1-4 1,679. The sum of ,13,341, being 10 per cent, of the net profits, has been added to the reserve fund. After due payment of the dividend on the 15 per cent, cumulative ordinary 5 04 A MODERN ARITHMETIC shares, the directors propose a dividend of 15 per cent, per annum on the A shares, leaving ,1838 to be carried forward. Find the total share capital (cumulative ordinary together with the A shares). Many attempts have been made at reducing the National Debt and interest thereon, as shown by the following exercises : 30. In 1822, ,152,422,143 Five per cents, was converted into Four %. ,2,794,276 was paid off to those people unwilling to accept the reduction. ;ioo of the 5 % was, however, converted into ,105 of the 4 %. Find the yearly saving to the country. 31. Find the yearly saving on the conversion of ,1,153,846 Bank of Ireland 5 per cents, into 4 per cents. 32. In 1824, 5 per cents, issued in 1797, were converted into 3 per cents. ,133. 6s. %d. 3 per cents, was given for each ;ioo in 5 per cents. The amount of 5 % stock was ,1,013,668. The amount held by dissentients .41,011. Find the amount of interest saved. 33. In 1830, 4 per cent, annuities were exchanged for .100 Three and a half per cents., or for ,70 Five per cents. The amount of the 4% annuities was .153,671,091. The amount held by dissentients was ,2,880,915 (paid off). The amount converted into 3^ % was ,150,1 19,609. Find the yearly saving. 34. What was the effective rate per cent, of a loan of 36 millions of pounds, issued in 1815, when, for ,100 sterling, ,174 stock at 3 might be obtained together with ;io stock at 4 % ? 35. Draw graphs showing the fluctuations in the highest, and in the lowest prices of Consols : (a) From 1838-1847 inclusive ; (b) from 1848-1857 inclusive ; 1858-1867 ( rf ) 1868-1877 using the data in the accompanying table : STOCKS AND SHARES 505 Consols. Consols. Highest. Lowest. Highest. Lowest. 1838 95* 9<> 1858 98| 931- 1839 93$ 89l 1859 97| 88| 1840 93l 85! 1860 95i 92i 1841 9oi 871 1861 94* 89 1842 94 88 1862 94| 9o| 1843 97 92 y 1863 94 90 1844 lOlf 96^ 1864 92 87i 1845 ioo| 9iiJ 1865 94 86f 1846 97l 94 1866 9! 4| 1847 93S 78| 1867 96i 8 9 | 1848 90 80 1868 96 9*1 1849 98| 88J 1869 94| 9 i| 1850 98g 942 1870 94i 88* 1851 99^ 95$ 1871 94 9i| 1852 102 95-H- 1872 93 9iJ 1853 101 9o| 1873 94 9if 1854 95-8- 855 1874 93t 9i* 1855 93l 86* 1875 951 9if 1856 95 851 1876 97i 93| 1857 94* 86| 1877 97g 93 70. STATISTICS. Averages. The term arithmetic average or mean has already been employed many times, and is the name given to the result which is obtained when the sum of a number of quantities is divided by the number of such quantities. Two other means are occasionally employed, thus : Suppose there are 8 quan- tities, the numerical values of which are 3, 5, 4, 6, 3, 8, 2, 6. The arithmetic mean = sum -7- number = J- {3 + 5 + 4 + 6 + 3 + 8 + 2 + 6} = ^ = 4-625. The geometric mean = \/3 X5x4x6x3x8x2x6 5 o6 A MODERN ARITHMETIC The use of logarithms shows the value to be 4-23606, and generally, if n be the number of quantities, their geometric mean is the nth root of their product. Again, suppose the reciprocal numbers be employed, a mean can be obtained as follows : The reciprocal of the arithmetic mean of the reciprocals of the The result, or harmonic mean = 2-075 As theoretical quantities, the meanings of the above results are clear enough, but in common parlance the word average is used in a very wide sense. The following examples afford material for the calculation of different means and also of various other averages. ,XAM Felt ham PLE I. JL lie W run as below. "eK-aay trains oeiween waierit What is the average duration 10 ana of the journey 1 W'LOO. FELT. W'LOO. FELT. FELT. W'LOO. FELT. W'LOO. 12. 10 a.m. 12.57 3-45 ?'" 4-35 4.45 a.tn. 5-46 4.II P .r a. 4.48 6.30 6. i 7.19 4- 7 4.25 4-49 5-15 5-51 6.46 6.48 7-44 5.II 5-30 6.20 7-15 8. 9 4-55 7-53 8.31 6. 8 649 7-30 8.17 5- 5 549 8.18 8.52 6.21 7- o 8.10 8.51 5.40 6. 5 8.31 9.19 6.50 7.40 9.15 10. 4 5-47 6.26 9- 3 9-34 7.29 8.10 9-30 10.12 6. 8 6.55 9.27 10.17 7-35 8.25 10. 5 10.55 6.40 7.14 9.42 10.30 8. 8 8.45 10.50 11.26 7.10 7-58 10.23 11. 6 8.20 9.11 11.20 12.12 745 8.25 10.30 11.23 9. 2 9-44 11.50 12.30 7.50 8.40 II. 10 9.l6 10. 7 1 2. 1 5 p.m. i- 5 8.10 8.50 11.52 12.35 9.58 10.34 12.40 1.20 8-35 9.24 12-53 p.m. IO. O 10.52 I.I5 1.52 9.10 9.48 1. 18 2. 7 ii. 6 11.47 1.30 2.24 9.58 10.50 1.39 2.12 11.15 12. 9 I.4O 2.20 10.35 II. 10 1.54 2.46 2.10 2. 5 1 1.50 12.18 3- 4 344 3-io 3-49 1 12.10 12.57 3-ii 4. 4 3.15 3-57 1 STATISTICS 507 The actual times in minutes are : 47, 5 6 49, 54, 47, 41, 49, 42, 50, 36, 52, 40, 50, 40, 37, 54, 40, 40, 39, 42, 50, 42, 50, 36, 44, 25, 39, 47, 34, 48, 40, 50, 40, 49, 38, 52, 35, 28, 47, 61, 57, 58, 38, 34, 48, 31, 50, 48, 43, 53, 51, 43, 38, 49, 33, 52, 40, 53, 37, 39, 5> 4i, 39, 5, 4', 5, 37, 5 1 , 42, 51, 3 6 , 5 2 , 4', 54- Analyse the results as below : Duration of journey, - 25 28 3i 33 34 35 36 37 38 minutes. No. of trains, I I i i 2 i 3 3 3 Totals, I 2 3 4 6 7 10 13 16 Duration of journey, - 39 40 4i 42 43 44 47 48 49 minutes. No. of trains, 4 7 4 4 2 i 4 3 4 Totals, 20 27 3i 35 37 38 42 45 49 Duration of journey, - 50 51 3 52 53 54 56 57 58 61 minutes. No. of trains, - "9 4 2 3 i i i i Totals, 58 61 65 6 7 70 71 72 73 74 The arithmetic average = { 25 + 28 + 3 1 + 33 + (2 x 34) + 35 + (3 x 36) + (3 x 37) + (3 x 38) + (4 x 39) + (7 x 40) + (4 x 41) + (4 x 42) + (2x43)+ 44 + (4x47) + (3x48) + (4x49) + (9 x 50) + (3 x 51) + (4 x 52) + (2 x 53) + (3 x 54) (74 being the number of trains) = (25 + 28 + 51 + 33 + 68 + 35 + 108 + 111 + 114+156 + 280+ 164+168 + 86 + 44+ 188+ 144+ 196+450 + 153 + 208 + 106 + 1 62 + 56 + 57 + 58 + 61) -7-74 = 44-7 nearly. 5 og A MODERN ARITHMETIC Thus the arithmetic average time is 44-7 min. No single train is timed exactly to this times, however, are not given to fractions of minutes. The calculation of the geometric mean is too long to he employed often in cases of this kind. For certain purposes it is more useful to take the mode than the arithmetic average, i.e. the time which most trains take, or the most com- mon time. 9 trains take 50 minutes on the journey, 7 take 40 min.; these are the times occurring most frequently, and the mode = 50 min. Perhaps more generally useful than either the arithmetic average, or the mode, is the median, a number calculated as follows : There are 74 train journeys ; arrange them as above in order of magnitude. It is seen from the line marked ' totals ' that there are 36 trains taking more than 44 minutes, and 37 trains taking less; also, that there are 35 trains taking less than 43 minutes, and 37 trains taking more, and the median lies between 43 and 44 minutes. If a graph be drawn it shows that practically there are two groups of trains, " fast" and " slow." EXAMPLE 2. The following table gives the times of the trains from London to Runcorn (Cheshire). Find the arithmetic average, the mode, and the median. Runcorn (Cheshire) from Euston, i8o| miles. EUSTON. 5.15a.m. 7.10 8.30 10.15 12. 10 p.m. 12.15 2.40 5-35 12. O RUNG. RUNG. EUSTON. 10. 11.46 1.10 I2.2Oa.m. 7.25 8.45 5.50 12. 1.40 2.45 IO.3O 3- T 5 4- 7 5-13 7-31 1 2. 24 p.m. 1.47 2.46 5-15 6.10 7.10 10. 5 8.17 4-34 6-39 8.50 10.45 7-35 3-50 STATISTICS 509 The times in minutes of the 19 journeys are : 285, 276, 280, 270, 237, 298, 291, 270, 497, 330, 275, 295, 285, 291, 263, 264, 256, 246, 495. Arrange the times in ascending order, thus: 237, 246, 256, 263, 264, 270, 270, 275, 276, 280, 285, 285, 291, 291, 295, 29 8 > 33, 495 497- The arithmetic average = -y-= 300-2 minutes. Note the great effect of the two very slow trains; without them the arithmetic mean would be 277-2 nearly. The Mode = 282 minutes. There are two journeys of 270 min., two of 285 and two of 291. The Median is the time such that the number of longer times is the same as the number of shorter times = loth in above arrangement = 280 min., and is not much affected by one or two abnormal times. EXAMPLE 3. The volumes and dimensions of ten cubes are measured^ and found to be as follows : volumes in c.c. 1-03, 1-16, 1-52, 1-03, 1-19, 1-73, 1-29, 1-73, 1-44, 1-23; edges in cm. i-oi, 1-05, 1-15, i-oi, 1-06, 1-20, 1-09, 1-20, 1-13, 1-07. Two people work out the average size of the cubes, one by dealing with the edges, the other with the volumes. Find the difference between the two averages. The arithmetic average of the edges = = 1-097 cm. volumes = 13-35 4- 10 = i -335 c.c. The edge of a cube of 1-335 c.c. = Vi-335 = i-ioi cm. (employ logarithms). The average length of the edge=i-ioi cm. deduced from the volume and = 1-097 cm. deduced from the edges themselves. Arrange the values of the edges and volumes in ascending order. Edges i-oi, i-oi, 1-05, 1-06, 1-07, 1-09, 1-13, 1-15, 1-20, 1-20 cm. Volumes 1-03, 1-03, 1-16, 1-19, 1-23, 1-29, 1-44, 1-52, 1-73, 1-73 c.c. 5 io A MODERN ARITHMETIC The median lies between the 5th and 6th values, i.e. be- tween 1-07 and 1-09, and has the same value whichever method has been employed. The geometric mean also would be the same for both methods (if the measurements were exact), and calculated from the edges = v the product of all the edges and the logarithm of the mean = ^{0-008632 + 0-021190 + 0-025306 + 0-029384 + 0-037427 + 0-053078 + 0-060698 + 0-158362} = T6-{-394077} =0-03940775 whence the mean or average edge= 1-095 cm - approx. Several quantities are used above in expressing an average. The ideal average should (1) Show a type if any type exists. (2) Give proper influence to extreme cases. (3) Be easily calculated, but not greatly affected by small errors in the data. ' Errors in Practical Work. Method of Mean Squares. In measuring a certain quantity, the following results are obtained: 4-02, 3-98, 4-00, 4-03, 4-01, 4-01, 3-99, 4-02, 3-97, 3'97- Here there are ten results and the arithmetic mean is 4-00. The differences between the quantities and the arithmetic mean are 0-02, -0-02, o, 0-03, o-oi, o-oi, -o-oi, 0-02, -0-03, -0-03. The sum of the squares of the differences = io- 4 {4 + 4 + + 9 + i + 1 + 1+ STATISTICS 511 The product of the number of readings, and the number reduced by Unity = 10 x 9-90. Divide this product into the sum of the squares of the differences ; extract the square root of the quotient ; it can be proved that the final answer gives the likely, or probable, error in taking the arithmetic mean as th^ true result. That is, the error / = \ 1 \ 42 X TO 4 _ IOX9 I-764 X IO~ 2 / * = 0-006 approx. .'. the likely error in taking 4-00 as the true value is 0-006. The true value probably lies between 4-006 and 3-994, and it is ctear that limits of this kind approach more closely as the number of the readings increases. EXERCISES XCIX. Find, from the data below, the average times of the given train journeys, stating (a) the arithmetic average, also (t>) the most common time, or 'mode,' and (c) the 'median' time. 1. London to Harrogate (Yorks) from King's Cross, 203 2. London to Harrogate. Another route from St. Pancras. 11J1A\^.3. ST. HARRO- HARRO- ST. KING'S HARRO- HARRO- KING'S PANCRAS. GATE. GATE. PANCRAS. CROSS. GATE. GATE. CROSS. 2.45 a.m. 10.10 7.30a.m. i- 5 5. 20 a. m. 10.42 7. Oa.m. 11.30 5-15 10.57 945 2.15 7-15 12.59 8. 5 i- 5 9-30 2.14 10.23 3-35 9-45 2-14 9- 5 i-55 11.30 3.56 12. 4-55 10. 2.30 10.10 2.20 12. 5-53 1 2. 27 p.m. 6. 20 10-35 4.19 10.27 3-55 1. 30 p.m. 6.30 2.40 8. 5 1.40? m. 5.42 10.57 4.10 2.15 8. 8 540 10.20 2.20 6.58 12.27 p.m. 5-30 3- o 8.38 6. 5 11.15 345 8. o T2.50 6.15 5- o 10. o 8.40 4.20 5-45 10.58 2.30 7- o 545 12. 5 10.25 5-30 6.15 12. 5 6. 10.45 12. 5-53 10.45 5-53 8. 5 2.40 11.30 8.20 8-35 3.20 1025 5.50 512 A MODERN ARITHMETIC 3. Leeds (Yorks) from King's Cross, i86 miles. 4. Leeds from St. PancraS, 196 miles. KING'S CR. LEEDS. LEEDS. KING'S CR. ST. PAN. LEEDS. LEEDS. ST PAN. 5. 1 5 a.m. 9.17 2.48 a.m. 8. 5 2.45 a.m. 9.12 12. 5a.m. 5-30 5-20 9.38 6.30 10.40 5-15 9.52 2-37 7- o 7-15 11.15 7-5 II.3O 8.30 1. 12 2-47 7-15 9-45 1.20 9- I- 5 9-30 1.22 3- o 7-35 10.35 1.30p.m. I.4O 3. O 5.12 6- 3 10.15 11.20 i '-33 i-55 3-55 4.10 10.30 11.30 12. 3- o 3.22 4-55 3.18 4. 2 6.10 7-30 8. o 10.40 2.20 7.11 2. Op.m. 5.30 1. 30 p.m. 5.30 6-35 10.40 3-45 7-45 2.55 7. O 2.15 6.37 8.30 i- 5 5-45 6.15 9-35 10.34 3- 5 5-30 8.30 9.25 3. o 4. o 7-37 8.57 10,20 II. 10 2.15 3-35 10.45 3-33 6.30 10.45 5- o 9.10 1 2. 45 p.m. 4-55 11.30 4-35 10.15 3-20 5-45 10.15 2.24 6. 20 7-15 11. 6 2.45 7-4<? 8.30 12.22 4. o 8. 5 9-3 1.40 6.22 10.20 12. 4. o 7.17 11,15 IO. O 4.20 5. Draw graphs in each of the cases 1-4, showing the number of trains which perform the journey in each time which 6. Tabulate the results above, grouping the times within 15 minute limits (thus, in (1) the first time is greater than 5 hrs. 15 min. and less than 5 hrs. 30 min.). Find the mode as calculated from the groups. 7. Find the (a) arithmetic, (b) geometric, and (c) harmonic means of the following numbers (the reciprocals are also stated). Numbers, - 22-1 23-5 24-2 23-9 24-3 Reciprocals, 0-045045 0-042533 0-041322 0-041841 0-041152 Numbers, - 25-1 23-2 22-8 24-0 23-6 Reciprocals, 0-039841 0-043103 0-043860 0-041667 0-042373 STATISTICS 513 8. The table below gives the annual premiums charged per 100 insurance by various Assurance companies : ANNUAL PREMIUMS FOR ASSURANCE, ETC. NAME OF OFFICE. (*) Age 21. .<> Age 30. to Age 40. <*) Age 50. W Age 60. s. d. s. d. s. d. s. d. s. d. Friends' Provident, - i 17 ii 280 340 497 6 13 ii General, - 200 2 9 10 354 4 12 8 6 18 o Gresham, i 19 8 290 3 5 8 4 H 3 7 6 5 Guardian, I 18 2 2 8 10 3 4 6 493 6 14 6 Law Life, 2 I 294 3 4 10 4 ii o 6 17 6 Law Union & Crown, i 18 6 284 340 4 9 10 6 16 o Legal and General, - 2 I 2 2 10 9. 35" 4 10 9 6 19 5 Life Assoc. of Scot. , i 19 8 2 IO O 354 4 13 4 724 L'pool&Lond. &G1., i 18 7 293 3 5 6 4 " 3 7 5 ii London & Lancashire, i 16 9 2 6 IO 324 4 6 10 6 18 ii London Assurance, - 208 296 3 4 ii 4 ii 5 7 2 ii Lond.,Edin. &Glas., i 19 ii 2 8 II 347 4 12 o 720 Marine & General, - i 19 o 2 8 10 35" 4 ii ii 700 Methodist & General, i 17 10 2 6 5 320 4 6 10 6 ii 9 Metropolitan, - 205 299 364 4 12 o 7 2 10 Mutual of Australasia, i 18 o 270 330 4 ii o 7 i o Mutual of New York, i 19 i 284 3 5 6 4 16 9 7 16 2 National Mutual, 209 284 337 496 662 Natl. Mutl. of Aust., i 17 7 268 3 i 6 472 6 18 10 National of Ireland, - i 19 8 2 8 7 343 4 ii 7 7 i 8 National Provident, - 203 2 IO 2 3 6 3 4 ii i 6 ii 10 New York, - i 19 3 289 345 4 17 o 7 19 10 Nth. Brit. & Mercan., i 19 i 2 9 10 3 6 i 4 ii ii 6 16 2 Northern, 2 I 2 290 348 4 10 10 6 17 4 Norwich Union, 235 2 II 9 366 4 12 5 720 Patriotic, I 19 I 288 3 4 5 4 10 4 6 16 4 Find the arithmetic average, the median, and the mode of the above rates. Also draw graphs, showing how these rates increase with the age of the person when he first insures. [In finding the mode in Ex. (a\ (b\ (c\ arrange the prices in groups, rising by 3^., i.e. rates 2. $s. od. to 2. $s. 2d. inclusive, 2. 3s, yi. to 2. 3s. $<t., etc. Arrange in groups, rising by 6d. in Ex. (d), and by is. $d. in (,).] J.M.A. 2 K 5I4 A MODERN ARITHMETIC 9. The following table represents the heights and weights of a number of boys between the ages of 13 and 15 : No. Height. Weight. No. Height. Weight. No. Height. Weight. Ft. Ins. St. Lbs. Ft. Ins. St. Lbs. Ft. Ins. St. Lbs. I 2 4 "4 4 10 6 of 5 7 27 28 5 i 4 9 5 nS 5 u 53 54 4 "4 4 5f 6 9f 4 94 3 4 5 5t 5 o U 29 30 5 if 4 9k 5 9 4 7i 56 5 44 4 10 6 24 1 5 31 4 10 8 04 5 o 32 5 i 6 84 5 i3i 58 4 9 5 04 5 5 6 4 7 4 10 6 7 33 4 8f 5 84 59 4 7 5 24 8 5 5 8 si 34 5 2 i 6 8i 60 4 9 5 9| 9 10 4 nf 5 i2i 6 ni 9 4 "4 4 ii 5 7 61 62 4 8| 4 84 ii ii 4 7 5 J 4 37 4 ii 6 of 63 5 54 7 10 12 5 3l 7 8| 38 5 of 6 2 64 4 8| 6 2j 13 4 7f 5 3 39 4 "4 6 34 65 5 4 7 2 14 7 8| 40 4 n4 5 7 66 4 9 5 04 15 5 o 6 o 4 ii 6 4i 67 4 7 5 o 16 4 9 5 6 42 4 44 4 n4 68 4 II 6 ii 17 5 24 7 74 43 5 3 6 74 69 4 i if 6 4i 18 4 8 5 3 44 6 i3i 70 4 8 4 44 19 4 ii 6 I2| 45 4 8| 6 of 4 8 4 44 20 5 i 6 9 46 5 2 7 4 72 4 71 4 10 21 4 ii 5 n 47 5 24 6 n 73 4 ii 6 5 22 4 8| 5 ii 48 5 4 7 44 74 4 34 4 ii 23 4 9l 5 8| 49 5 i4 6 10 75 c o 7 2 f 24 5 2 J 6 i 50 c o 76 4 Si 5 6 25 4 6 5 64 5 ! i 6 7 26 5 34 7 6 4 5 2 4 "4 7 3i Find () the median weight, also (b) the mode, and (^)"the arithmetic average. Find also the average weights of boys 4 ft. 6 in. or over, but not 4 ft. 7 in. ; 4 ft. 7 in. or over, but not 4 ft. 8 in., and so on. Draw ordinary and logarithmic graphs connecting these average weights with the height. 10. Draw charts, showing the fluctuation in the proportion of the total imports and exports of manufactured articles into and from the under-mentioned countries, which were respectively derived from, or exported to the United Kingdom, in each of the years from 1890 to 1901 inclusive : STATISTICS 515 (i) IMPORTS OF MANUFACTURES. PROPORTION (PER CENT.) DERIVED FROM THE UNITED KINGDOM. to (a) / L\ /-\ (ft) United States Years. Germany (Estimated). \?) France. '*/. Russia. Italy (Estimated). (Years ended 3oth June) (Estimated). 1890 47-1 38-8 23-3 40-3 4 8-4 1891 47-1 367 25-8 ... 47-3 1892 49-4 36-6 28-5 31-7 44-1 1893 49-2 38-1 26-8 33-4 44-6 1894 46-6 38-4 27-0 31-0 39'8 1895 46-5 39-9 22-8 29-5 44-5 1896 48-9 38-9 21-0 31-8 45-6 I8 9 7 48-0 36-6 21-2 26-3 42-9 l8 9 8 44-9 34-8 21-5 27-6 4o-3 1899 45-2 32-3 22-3 23-9 37-8 1900 40-3 29-7 19-9 22-8 42-2 1901 34-4 28-5 ... 17-6 36-5 (ii) EXPORTS OF MANUFACTURES. PROPORTION (PER CENT.) EXPORTED TO THE UNITED KINGDOM. Years. _ CO Germany (Estimated). to France. W. Russia. CO Italy (Estimated). (/) United States (Years ended 3oth June). 1890 10-9 26-3 2-7 9'9 1891 1 1-7 26-8 2-6 ... 1892 n-3 30-6 2-4 8-4 24-5 1893 ii-3 29-6 2-3 7-2 23-9 1894 12-3 2 9 -I 4-3 7-4 25-4 1895 II-2 30-2 4-0 67 30-5 1896 12-2 29-8 3-6 6-8 28-9 1897 13-0 30-3 3-3 7-4 30-5 1898 12-8 28-9 3-o 6-5 27-3 1899 12-2 30-9 3-6 7-i 26-0 I9OO II-8 29-5 3-2 7-5 22-4 I9OI 13-4 30-8 6-6 24-8 Find also (i) the arithmetic average and (ii) the median in each case. 5 i6 A MODERN ARITHMETIC 11. The following table shows percentage changes in the retail prices of certain articles of food of workmen's consumption, 1877-1901. (Price of each article in 1901 equals 100.) A. IN LONDON. 1 $3 O si 1 S I > "' 3 s 1 ?z -1 S "8 ^ H 1877 200-7 181-1 158-2 141-1 127.9 97-0 97-6 I2O-O 1 60-0 1878 1 80-0 169-8 121-4 141-1 124-0 96-0 94-9 I2O-O 140-0 1879 i3'-4 186-6 136-8. 128-0 126-0 96-0 82-4 I2O-O 120-0 1880 1 60-0 178-3 152-5 141-1 127-9 96-0 90-1 I2O-O I30-0 1881 148-6 163-8 148-3 I38-5 131-8 104-0 80-8 120-0 I30-0 1882 148-6 155-5 135-8 135-9 141-5 104-0 85-8 120-0 13O-O 1883 148-6 166-4 123-4 135-9 143-4 c.6-0 96-5 12O-O 13O-O 1884 125-7 134-9 "3-4 128-0 127.9 96-0 95-1 I2O-O 120-0 1885 102-9 134-0 118-2 122-8 104.7 96-0 85-8 I2O-O 80-0 1886 102-9 129-4 118-2 101-9 96-9 88-0 87,2 120-0 9O-O 1887 108-6 I35-I 108-5 95-4 89-8 72-0 96-3 120-0 80-0 1888 IOO-O 124-5 1 1 1-9 98-1 96.9 68-0 92-7 120-0 9O-O 1889 1 1 1-4 124-5 130-6 96-3 102-3 72-0 87-2 I2O-O 00-0 1890 IOO-O 114-7 130-6 96-3 103-1 86-0 92-7 IIO-O 80-0 1891 105.7 146-4 130-6 98-1 96-1 88-0 98-2 95-0 80-0 1892 117-1 122-6 128-9 99-i 96-1 92-0 101-8 95-0 90-0 1893 85-7 I2O-O 104-5 97-2 90-6 1 06-0 103-7 95-0 80-0 1894 85-7 124-9 Q8-3 99-1 93-8 90-0 95-4 90-0 80-0 1895 77-1 133-8 91-8 96-3 94-5 82-0 88-1 90-0 60-0 1896 85-7 95-3 95-3 96-3 89-1 70-0 95-4 90-0 80-0 1897 102-9 98-3 115-2 97-2 92-2 90-0 95-4 90-0 60-0 1898 122-9 125-3 130-6 96-3 92-2 92-0 96-3 90-0 6c-o 1899 91.4 101-9 99-3 98-1 94-5 82-0 ico-9 90-0 80-0 1900 97-1 91-5 107-5 IOO-O 977 90-0 IOO-O IOO-O 80-0 1901 IOO-O IOO-O IOO-O 100-0 ICO-O ICO-O IOO-O IOO-O IOO-O Draw charts showing 'the changes ; find also (i) the mode and (ii) median in each case. (In finding the mode, arrange the prices in suitable groups 87-89, 90-92 inclusive, and so on.) STATISTICS 517 12. The table below gives the minimum average rate of discount charged by the Bank of England for different years, beginning with 1854, and extending to 1902 inclusive : 5, 4l, 52. 6, 3l 5 2f , 4i, 5 J *\> 7l 5 4l, 4f, 3l J 3l, 2|, , 2l, 2}, 3] ; 3 J, 2 |, 2 | ; 2 f , 3 J, 4, 3A, 2*& ; 3, 3, 3i 3k 3l 5 4, 3i, 35. 2 i 32\> J 25, 2, 2*, 2*, 3 J. ; 3 J, 3 |jj, 311, 3 -j. Find () the changes in the quinquennial average (from 1855 to 1859, 1860 to 1864, etc., up to 1895-1899) ; find also (b) the arith- metic average, (r) the mode (dividing into groups of 0-5) and (d) the median. 13. (i) The average prices per ,100 of the 3% Consolidated Stock of the Public Funds of the United Kingdom are given below. Find (a) the quinquennial averages and (d) the arithmetic average during the whole time taken, also (c) the mode (dividing into suitable groups) and (d) the median. Year - 1855 1856 1857 1858 1859 1860 1861 1862 1863 1864 Price 90? 2 931 9ii 96* 951 94 9il 93 92| 901 Year 1865 1866 1867 1868 1869 1870 1871 1872 1873 1874 Price 89l 88 93 931 92|- 921 92| 92-i 921 92^ Year - 1875 1876 1877 1878 1879 1880 1881 1882 1883 1884 Price 93! 95 95-1 95 A 971 98| IOO ioo lOlf 101 (ii) Find similar averages to (i) for the value of the British coinage at the London Mint between 1884 and 1904 inclusive. Year. Amount. Year. Amount. Year. Amount. 1884 2,321,000 1891 6,723,600 l8 9 8 5,780,600 1885 2,973,500 1892 13.907,800 l8 99 9,010,900 1886 None. 1893 9,266,300 1900 13,103,700 1887 1,908,700 1894 5,678,100 1901 2,6O4,OOO 1888 2,033,000 1895 3,811,200 1902 7,119,000 1889 7,500,800 1896 4,808,900 1903 9,928,000 1890 7,680,200 1897 1,778,300 1904 II,042,OOO 5 i8 A MODERN ARITHMETIC 14. The following table shows the average annual price of bar silver, per ounce standard, in each of the undermentioned years : Year - 1835 1836 1837 1838 1839 1840 1841 1842 1843 1844 Price in \ pence } 591* 60 59T 9 TT 59* 6of 6o| 6o T V 59rV 59r 3 ir 59* Year - 1845 1846 1847 1848 1849 1850 1851 1852 1853 1854 Price in \ pence / 591- 59& S9H 59* 591 6o T V 61 6o\ 60^ 6U Year - i855 1856 1857 1858 1859 1860 1861 1862 1863 1864 Price in \ pence / 6i& 6iA 6if Sly 5 * 62 T V 6iH 6oH 6iA 6if 6if Year - 1865 1866 1867 1868 1869 1870 1871 1872 1873 1874 Price in \ pence / 6i T V 6ii 6oA 60* 6o T V 6o r % 60^ 6o T <V 59i SA Year - 1875 1876 1877 1878 1879 1880 1881 1882 1883 1884 Price in I pence / 5<* 52f S4H 52 T 9 TT 5il 52! 57H 5if SOi 9 * 5of Year - 1885 1886 1887 1888 1889 1890 1891 1892 1893 1894 Price in \ pence / 4 8f 451 44t 42t 42H 47H 40^ 39it 35f 281f Year - i895 1896 1897 1898 1899 Price in \ pence / 291 30J 27T 9 C 26H 27A Find (a) the quinquennial averages, also () their general arithmetic average through the whole range, (c) the arithmetic average between the years 1835-1874 inclusive, (d) the mode (when arranged in suitable groups) and (e) the median. 15. A number of consecutive daily barometric readings were as follows : 29-64, 29-81, 29-91, 29-87, 29-87, 30-04, 30-11, 29-99, 30-21, (missing), 30-23, 30-26 in. The arithmetic average is known to be 30-01, with a possible error of 0-005. What is the missing reading? What errors are possible? Find also a probable value by means of a graph. STATISTICS 519 '16. Calculate the (a) mode, (b) median and (c) arithmetic average of the annual rainfall in the Thames basin during the period 1883-1902. (In calculating the mode, arrange in groups ranging within i inch, i.e. 26-27 in., 27-28 in., etc.) Year 1883 1884 1885 1886 1887 1888 1889 1890 1891 1892 Rainfall \ in inches/ 28-4 22-9 29-1 3I-I 21-3 28-4 2 5 -6 22-8 33-3 23-0 Year 1893 1894 1895 1896 I8 97 1898 1899 1900 I9OI I9O2 Rainfall ] in inches/ 22-1 32-3 26-3 2 5 .8 27-8 22-1 24-8 27-9 23-5 21-9 17. A number of cubes are weighed first in air, then in water, and the loss of weight found to be 2-537 gr. When placed in a row, face to face, the total length is found to be 6-333 cm. State (a) what you think the number of cubes probably is, also (b} the average edge. 18. Calculate probable values for the following : (a) A length, different measurements of which are 32-03, 32-04, 32-02, 32-05, 32-01, 32-06, 32-02. (b) The wave length of a light from the following readings : 4,r=i-8o, 1-81, 1-81, 1-79, 1-78 mm., where the wave length =x(-\ and a=ii2-o cm., while readings for c are 0-092, 0-091, 0-093, 0-093, - 9 2 cm - (c) The number of centimetres a body falls through from rest in the first second of its motion, from the following readings : 490, 491, 489, 489, 491, 492, 498. 520 A MODERN ARITHMETIC 19. The table below gives the average yearly prices of wheat in the United Kingdom, with remarks as to cases where the price seems exceptional : Year. Average Price Per Imperial Qr. Year. Average Price Per Imperial Qr. Year. Average Price Per Imperial Qr. s. d. j. d. J. d. 1870 46 II 1881 45 4 1892 30 3 I87I 56 8 T882 45 i '893 26 4 1872 57 o I88 3 4i 7 1894 22 10 1873 58 8 l88 4 35 8 I8 95 23 I 1874 55 9 1885 32 10 l8 9 6 26 2 1875 45 2 1886 31 o 1897 30 2 1876 46 2 1887 32 6 1898 34 o 1877 56 9 1888 31 10 1899 25 8 l8 7 8 46 5 1889 29 9 I9OO 26 ii 1879 43 10 1890 31 ii I9OI 26 9 1880 44 4 1891 37 o 1902 28 i Remarks. 1876 Bad harvest. 1891 Failure of rye harvest in Russia. 1897-1898 Wheat crops generally short. 1898 'Leiter corner.' 1891-1895 Series of abundant harvests. Draw a chart showing the fluctuation in price ; calculate also the arithmetic average for (a) 10 years 1872-1381, (6) 1882-1891, (c) 1892-1901, and express the average prices as percentages of the price in 1871. Compare also the fall in price in wheat with the fall in com- modities generally, as shown by the following table : Price of Commodities 1871 1872-1881 1882-1891 1892-1901 As % of Prices in 1871 100 104-8 84-7 75-8 STATISTICS 521 Permutations and Combinations. EXAMPLE i. Find the number of ways in which the vowels a, e, i, o, u may be arranged in groups of $. Here we may say there are 3 vacant places to be filled by the vowels D O D- For the ist place we have 5 choices #, <?, /, <$>, u ; .'. it may be filled up in 5 ways ; for the 2nd place we have 4 choices, each one of which may be associated with each of the ist ; for the 3rd place, we have 3 choices ; .'. the total number of ways is 5 x 4 x 3 or 60. EXAMPLE 2. In how many ways could all the letters of the word ' vowels ' be arranged ? Here there are 6 different letters, and the number of arrange- ments is 6 x 5 x 4 x 3 x 2 x i = 720. EXAMPLE 3. In how many ways could 4 three-quarter backs be picked from 7 possible players^ disregarding any question as to their positions ? If the position of the men on the field were taken into account, say Left Inside left Inside right Right, the number of possible arrangements would be 7x6x5x4, but any one choice of four players could be arranged in 4x3x2x1 ways. r ,. 7x6x5x4 The number of choices = - - ^ r 4x3x2x1 The four players could therefore be chosen in 35 ways. Probabilities. Important applications of averages are the calculation of probabilities, chances, expectations, etc. From the examination of many mortality returns it is found that out of a large number of persons alive, say at 55 years, 522 A MODERN ARITHMETIC a certain number would be alive, say at 60 years. Thus, in one case, out of 424,677 persons alive at fifty-five, 365,011 were alive at sixty. The probability that a person alive at 55 would be alive at 60 is 3 5 OII ? an( j j t i s evident that " probability " is really 424677 an " average probability." Similarly, if there is a large number of equally possible events N, and in n of them a certain result happens, then the probability or chance of the result happening is . The "odds" that the event happens is No. of favourable cases n No. of unfavourable cases N - n Thus, if a coin be thrown up, the chance that it falls with ' heads ' uppermost is \. If a die is thrown the chance that a 3 turns up is \ and so on, and the "odds" against it, 5 to i. EXAMPLE i. Two dice are thrown together ; what is the chance that the throw is 9 ? The total number of possible throws is 6 x 6 or 36. In these throws 9 can occur by the following arrangements : i.e. in 4 ways. The number of favourable cases = 4. Total number = 36. .'. the chance is or . STATISTICS 523 EXAMPLE 2. Find the chance that a man throws a 5 once in 3 throws of a single die, stopping directly the 5 has been thrown. The chance of turning up a 5 in the ist throw = \. having a 2nd throw = chance of missing in the ist throw. .'. chance of turning up a 5 in the 2nd throw = | \. The chance of having a 3rd throw = chance of missing in ist two throws = (f-) 2 . The chance of turning up a 5 in the 3rd throw = ( J) 2 . \. Total _ 91 2T6"' EXAMPLE 3. A purse contains 7 similarly marked counters : one marked to represent ictf., two others to represent 55-., the rest to represent is. If a person be allowed to pick a counter at random, and thereby becomes entitled to the sum of money marked on the counter, find the expectation value of the drawing. Here the chance of drawing a i os. counter = \ ; .'. expectation = if- shillings. - - _ 2 _ 2X5 5*' 5) T ?> '~T~ " .*. total expectation value is = + = 3 f- EXAMPLE 4. The followi?ig are turf quotations for a horse race, taken from a daily paper : 2 to i against horse A. 9 to 2 B. 10 to i C. 100 to 9 D. 100 to 8 ,, E. 100 to 7 F. 33 to I G - Assuming that the above represent the chances of the respective horses winning, find the expectation of a person who bets j^\ on each horse. 524 A MODERN ARITHMETIC Chance of A winning is ~- = , since odds are 2 : i against ; .*. expectation = ^ of 2. So for the others, whence the total expectation - {* + A * 1 + TT x 10 + jgg + igf + is? + if} = ;6. 2J. I0</., for which the person has paid ^{1 + 1 + 1 + 1 + 1 + 1 + 1}= ^7. EXERCISES C. 1. In how many ways could you arrange the letters of the words (a) Four, (b) W 7 hile, (c) Result ? 2. In how many ways could you arrange three letters chosen from the words (a) Ways, (b} Newmark, (c) Chosen ? 3. In how many ways could a group of four letters be chosen (not arranged) from the English Alphabet ? 4. Two dice are thrown ; find the chance that the throw is (a) 10 exactly, (b} less than 10, (c) greater than 10. 5. A cricket eleven is to be picked from fifteen players ; in how many ways can this be done ? 6. A football fifteen has to be chosen from 10 eligible forwards (8 forwards to be chosen). 4 half-backs (2 to be chosen). 6 three quarters (4 to be chosen). 3 full backs (i to be chosen). In how many ways might the team be made up ? 7. Find the chance that if three cards be taken in succession from the same pack, they shall be king, queen and knave of the same suit ? Expectation of Life. The table below gives the expectation of life at different ages. The numbers in the first three columns are the results STATISTICS 525 of observations, those in columns four and five are obtained by calculation founded on the death-rates of 1871-80 : Age. Of 1,000,000 Born, the Number Surviving at the end of each Year of Life. Mean After-lifetime (Expectation of Life). Male. Female. Male. Female. 12 703,595 732,697 45-96 48-13 17 690,746 718,993 41-76 44-00 18 687,507 715,622 40-96 43-21 19 683,941 711,946 40-17 42-43 26 651,998 679,822 34-96 37-26 27 646,757 674,661 34-24 36-54 33 611,827 641,045 30-01 32-30 36 592,107 622,554 27-96 30-21 37 585,167 616,144 27-29 29-52 40 563,077 596,113 25-30 27-46 4i 555,254 589,167 24-65 26-78 48 495,761 537,043 20- 1 8 22-03 49 684,794 529,048 19-55 21-36 50 476,980 520,901 18-93 20-68 5i 467,254 512,607 18-31 2O-OI 55 424,677 477,440 '5-95 17-33 56 413,351 467,443 15-37 16-69 60 365,011 422,835 13-14 14-24 61 62 352,071 338,820 410,477 397,644 12-60 12-07 I3-65 I3-08 63 325,256 3 8 4,3I9 11-56 12-51 64 3",368 370,495 11-05 II-96 70 222,056 277,225 8-27 8-95 7i 206,539 26o,2O7 7-85 8-50 72 190,971 242,934 7-45 8-07 73 175,449 225,497 7-07 7-65 74 75 160,074 144,960 2O8,OO3 190,566 6-70 6-34 7-25 6-87 76 130,227 173,316 6-00 6-51 77 115,986 156,392 5-68 6-16 78 102,359 139,927 5-37 5-82 79 89,449 124,065 5-07 5-50 80 77,354 108,935 4-79 5-20 81 66,153 94,662 4-5i 4-90 82 55,842 8l,305 4-26 4-63 83 46,489 68,966 4-01 4-37 84 38,132 57,723 3-58 4-12 5 26 A MODERN ARITHMETIC EXAMPLE i. Employ tfie table on p. 525 to evaluate the value of life annuities at different ages ; for example, to find the value of a life annuity of i at the age of 50, the insurer being male. From the table the expectation of life is 18-93 years, and the value of the annuity is very nearly the present value of 19 yearly payments of i. Assume 3 % per annum, Compound Interest, payable yearly. The present value in = i + - + 03 v * ~ 0/ ;( . . .19 But, " (cf. tables); ^ = 0-97087' the value of -/Ti annuity 0-42071 - = ;i4. 155. approx., 1-0-97087 0-02913 while ;ioo would buy a yearly annuity of 6 '" 8 ^ 6 - I55 - ^ approx - Similarly, the tables may be employed in calculating the value of a Reversion. Consider the following : EXAMPLE 2. What is the value of a Reversion to ^2750 on the death of a lady, aged 77 years? (Compound Interest, ai 3 % per annum.} When the lady dies, the sum of ^2750 will be in the pos session of the holder of the reversion. The expectation to the nearest year of after life in the case of a lady, aged 77, is 6 years. STATISTICS 527 The present value is therefore required of 2750, due in 5 years. .'. the present value = 2303. is. (to the nearest is.). EXAMPLE 3. An Insurance Company charges a certain Premium to a male client who begins insuring at 26 years, the :alculation employed in fixing the premium is based on money earning interest at 2^ % per annum. What is the price of the Premium (disregarding expenses of management, etc.} on each 100 policy ? The expectation of life, at 26, for a male is 35 years. Now i per annum will amount in 35 years to "54-9282 (see tables) "4%- .'. the premium per i EXAMPLE 4. If of every 1,000,000 male persons born, 77354 mrvive at 80 years of age, and if of every 1,000,000 male persons Tjorn 66153 survive at 81 years of age, and if the expectation rf life at 8 1 years is given as 4-51 y ear S, find tlie expectation of life at 80 years. Out of 77354 people 11201 (i.e. 77354-66153) die during the ensuing year. Hence the probability of dying within the pear is - = 0-145 approx., and the probability of living / / O T" until the end of the year = (1-0-145) = 0-855. For the men who live through the year the expectation is 4.-5I years at the end of the year, or 5-51 years at the beginning of the year. For the men who die, assume that the average life was 0-5 yr.,. whence the expectation at 80 = 4-79 years. 5 28 A MODERN ARITHMETIC EXERCISES 01. Use the table on page 525 to find, by calculation, the value of the immediate annuity that may be purchased by the payment of ;ioo at the following ages and rates of interest, the purchaser being (a) male, (V) female. (Take expectation of life to the nearest year, and state answer to nearest penny.) 1. Age 55, rate of interest 2^%. ^* 55 55 55 55 3 To* 3. 60, 3%. 4. 40, 2|%. 5. ,5 75 55 J5 4 % earest Employ the mortality table on page 525 to find the insurance premium on 100 policy (without profits) at 2^% per cent, per annum, Compound Interest. 6. When the insurer is male, aged 12 years. 7. 8. 9. 10. 11. 12. female, 19 33 36 48 17 13. Of 1,000,000 persons born the numbers surviving (census returns 1871-1880) at the end of each of the given years of life, are as follows : Age. Male. Female. 50 51 476,880 520,901 467,254 512,607 52 457,022 504,188 53 446,510 495^45 54 4355729 486,973 STATISTICS 529 The expectation of life at 54 is 16-53 years for a male. 17-98 female. Find the expectations at (a) 53, (b) 52, (c) 51, (d) 50 for a male, (*) 53, (/) 52, (,) 5 1 , (A) So for a female. 14. Find the values of Reversions to the following : (a) ,800, on death of lady, aged 72 years. (b) ^1200, man, 83 (c) ^1500, 79 (d) 7575, lady, 64 0) ^1250, 70 (Take expectation of life to nearest year.) Assume Compound Interest at 3% per annum. REVISION EXERCISES Oil. Miscellaneous : Mainly from Examination Papers. A 1. Explain the rule for finding the G.C.M. of two numbers, also any methods with which you are acquainted for simplifying the operations. Find the G.C.M. of 622932 and 2087664. Also the L.C.M. of 3663, 16280 and 9768. 2. (i) Reduce 0-63 of i. os. 4^. to the decimal of 0-5625 of 4- ^s. (ii) Find the value of 0-18637 of 4, true to the nearest penny. 3. The sides of a rectangular piece of paper are measured both in centimetres and in inches, and the length of a metre in inches is determined from each set of measurements. Find, to 2 decimal places, the average of the two results if the length of one side of the rectangle is 16 inches or 40-6 cm., and that of the other 1275 inches or 32-5 cm. J.M.A. 2L 530 A MODERN ARITHMETIC 4. An observatory building consists of a cylindrical base of 12 feet radius and 13 feet high, surmounted by a hemisphere of the same radius. Find the cost of covering the outside with water- proofing material at 4s. 6d. per square yard. 5. Two men begin at noon to fill a cistern holding 200 gallons. The first empties a three-gallon cask into the cistern at the end of every two minutes, while the second empties a two-gallon cask into it at the end of every three minutes. When will the cistern be full? 6. A man, who can row three miles per hour in still water, rows to a place 9 miles down stream and back again. How long does he take if the stream flow at the rate of 2 miles per hour ? 7. Find the rate per cent, at which a customer pays interest, in buying the following on credit terms : " We send a cycle on receipt of is. deposit and payment of is. per week for 89 weeks, making a total of ^4. los. in all, or our net price is only ,4." 8. The weight of 1000 sovereigns is 7-988 kilo., that of 1000 twenty-mark pieces is 7-962. If ^i = 20-43 marks at the mint par of exchange, find the fineness of the twenty-mark piece. 9. A capitalist has ,27,900 which he invests, part in 4 per cent, debenture stock at 114$, and part in 3 per cent, stock at 96|. If the resulting income be ^873, find how much is invested in each stock, allowing as brokerage, 2s. 6d. for each ^100 stock and ^3 for stamp fees on the total amount invested. 10. AC is a straight railway and B a place 5 miles from the nearest point C of the railway. Suppose goods to be sent from A to a siding 8 by rail and thence by road to B, S being x miles from C. The cost per mile being i shilling by rail, and the cost per mile by road 2 shillings and the distance AC 12 miles, give an expression for the total cost of carriage. Draw a graph showing the cost of carriage for different values of x, and determine where the siding S must be placed in order to make the cost as small as possible. For the graph represent r shilling and also i mile by 2 centimetres. REVISION EXERCISES 531 B 1. Obtain the least perfect cube which is exactly divisible by 42, 56, 154, 231 and 484. (Factorize the numbers.) 2. A cubic foot of water weighs approximately 62^ Ib. A litre of water weighs a kilogram. An inch = 2-54 cm. From these data calculate to one decimal place the number of grams in i Ib. 3. How far does a roller, width 5 ft. 3 inches, have to be pulled in rolling a rectangular field of area 7^ acres ? Allow 10 per cent, extra for turning, and assume that the whole field can be covered without one crease overlapping another. 4. The side wall of a lean-to shed slopes from a height of 5 feet to a height of 20 feet, and 17 upright slips of wood are fastened to the wall at equal intervals, each reaching from the ground to the top of the wall. Find the total length of the wood. 5. Three persons, A, B, C, enter into partnership with sums of ^1800, ^1500 and ^1400 respectively. A withdraws ^600 at the end of 3 months, and the deficit in the capital is made up by B subscribing ,300 and C ,300. The profits at the end of a year's business amount to ^840 ; how should it be divided ? 6. A bicycle was sold at a loss of 40 per cent, on the cost price, and a second bicycle was bought with the proceeds when 5 guineas had been added. This second bicycle was sold at a loss of 35 per cent., and a third bicycle was bought with the proceeds after the addition of ^5. 15^. ^d. This third bicycle cost 16 ; what was the price of the first ? 7. A hollow ball, of external diameter 6 centimetres and of thickness half a centimetre, weighs 162 grams when it is filled with liquid. Find to the nearest hundredth of a gram the weight of a cubic centimetre of the liquid, a cubic centimetre of the substance of the ball weighing 2 grams. (If a sphere is inscribed in a cube, then volume of sphere : volume of cube : : n : 21.) 8. Tithe rent charge in 1901 is of such value that every nominal ,100 is really worth only 66. los. ofcd. Find the amount of tithe rent charge which is worth \. 14^. $d. less than its nominal value. 53 2 A MODERN ARITHMETIC 9. A member of a building society obtains a loan of ^400 for the purchase of a house, and agrees to repay it with compound interest at the rate of 5 per cent, in five equal annual instalments, the first payment to be made at the end of a year. What should each instalment be ? 10. In an experiment with a screw jack the following results were obtained : Load W lifted in pounds Power P applied in pounds. IOO 12-6 120 I3-8 140 157 1 60 17-6 .ISO 19-6 2OO 2I-5 Plot these observations, draw a fair line through them, and from your diagram read off the values of a and b which satisfy the ec l uation 01. In subtraction how do you get over the difficulty of taking a greater digit from a less ? Illustrate your answer by taking 863 from 952. By what factor less than 1000 must 7683 be multiplied so that the last three figures of the product may be 768 ? 2. Find the area of the walls of a room 23 ft. 8 in. long, 18 ft. 4 in. wide and 12 ft. high, and the expense of covering them with paper 3 ft. 6 in. wide costing 43. 6d. per dozen yards. 3. Obtain the square root of 11-0405067, exact to four places of decimals. 4. A sum of ^371 is due to me at the end of 18 months ; what sum of money paid down now would discharge the debt, reckoning simple interest at 4 per cent? REVISION EXERCISES 533 5. Explain the following extract with reference to the British coinage : Denomination. Standard Weight. Least Current Weight. Remedy of Weight. Gold. Sovereign Grains. 123-27447 Grains. 122-500 Grains. O-2OOOO Silver. Shilling 87-27272 0-578 A mining company extracted 19744 tons of gold ore, yielding 2275 oz. of gold. What number of sovereigns would the mint supply from this ? [Mint gold is of}.} fineness.] 6. The squares of the times of revolution of the planets of the Solar system are as the cubes of their distances from the Sun. Find, to the nearest day, the time Jupiter takes to go round the Sun, given that its distance from the Sun =5-2 times the Earth's distance. 7. A person bought the materials of an old house for upon condition that they should be removed within 12 days under a penalty of ^i a day for every day over the 12 days. He paid ^3. IDS. a day for labour, and having sold the materials for ^214. loj., he found that he had made a profit of ^30. How many days did he take to remove the materials ? 8. A shallow tinplate dish is in the form of ^a frustum of a cone. Its depth is 3 inches, its base a circle of 6-inch radius, its upper margin a circle of 8-inch radius. Find its total surface. 9. If pure gold be worth ,3. 17.9. 6d. per oz. and silver be worth is. %d. per oz., what will be the value of an alloy consisting of 3 parts of the former to 5 of the latter ? If this alloy be sold at ,1. 13^. 7.\d. per oz., what is the gain per cent. ? 10. The distance, s feet, of a moving point from a fixed point at time, / seconds, is given in the form s=io + 8t+6t 2 . Find 534 A MODERN ARITHMETIC the average velocity during the first 6 seconds. Find also the velocity at the time /=3 (i) by drawing the distance-time curve, (2) by determining the average velocity during the interval between ^=3 and /=3 + *, when x is very small indeed. D 1. The numbers 4242, 2903, when divided by a certain number which contains three digits, leave the same remainder ; find that remainder. 2. Of the children attending school in Switzerland, 71-7 per cent, speak German only, 22-4 speak French only, 4-5 speak Italian only, and the remainder consists of 5,838 children who speak only Romansch ; find the total number of children attending school. 3. Given that one metre = 39-371 inches, and i kilogram = 2-2046 lb., and if i cubic metre of water weigh 1,000 kilograms, find the weight of a cubic foot of water in lb., correct to the nearest tenth of an ounce. Use contracted methods of multiplica- tion and division. 4. Three persons contribute sums of ,250, ^500 and ,750 respectively towards a venture, on the understanding that the profits shall be divided in such a way that the rate of interest which each receives shall be in proportion to the amount of his contribution. If the profits for a year amount to ^245, how much will each of them receive? 5. Find the cost of discounting on Sept. i, 1904, a bill of ,580. I2j\, maturing on Oct. 18, 1904. (Mercantile Discount at the rate of 4 per cent, per annum.) 6. Express 0-296 as an infinite geometrical series, find its sum, and show that it is equal to (o-6) 3 . 7. On a map is shown a triangular piece of land bounded by three roads which measure 4 in., 4 in., 3 in. The whole map is 1 8 inches long and 12 inches wide, and represents 216 sq. miles. What area does the triangle represent ? Use a carefully drawn figure to find the area of the triangle. REVISION EXERCISES 535 8. The radii of the circular faces of a frustum of a right cone are 12 and 8 feet, and the area of its curved surface is 20^24.1 square feet ; find the thickness of the frustum. 9. By buying 3 per cent, consols at a certain price I find I obtain 3^ per cent, for my money, and derive a net income there- from, after paying an income-tax of 6d. in the ^, of ,421. 4.?. Find the amount of stock and the price at which I bought it. 10. The gas-service pipe to a house 75 feet from the main is in. in diameter ; for how many burners, each taking five cubic feet of gas per hour, will this serve ? The number of cubic feet per hour delivered by a pipe on that main is 1000 -, where 0-45 L d is the diameter of the pipe in inches, and L is the length of the pipe in yards. E 1. (a) Add together 3&, i, 26f, 5f, 20^. (b) A room is 45 feet 6 inches long and 23 feet 10 inches broad. Could it be covered exactly with tiles 13 inches square? How many would be required ? 2. Find the prime factors of 1066, 1815 and 1896; and hence write down the factors of their L.C.M. 3. Reduce 35469099 cub. in. to cub. ft. by short division, explaining how the remainder is obtained, and the reason of the rule. 4. A customer obtains from his banker the loan of ^1250 on June 2nd ; he pays ^450 on July 2nd ; ^600 on October ist, and finally pays in ^800 on November ist. How does the customer stand with regard to the bank on December 3ist, assuming the bank charges 5 per cent, per annum interest on all sums that the customer owes it, but allows the customer interest at 2^ per cent, per annum on all sums standing to the credit of the customer. 536 A MODERN ARITHMETIC 5. Find the change in a man's income, occasioned by selling ,10,000 Two and three quarters per cent, consols at 101^-, and investing the proceeds in Corporation of London 3^ per cent. Debentures at 100. Brokerage ^ per cent, on Consols, i per cent, on the Debentures. 6. A silver bowl weighs 67^ oz. of 480 grains each. What is its weight in French kilograms, etc., to the nearest gram [a gram = 15-45 grains]. 7. The times of departure from Euston, and arrival at Birmingham, of passenger trains (starting between 6 a.m. and 6 p.m.) are as follows : Euston. Birmingham. a.m. 7.10 10.12 7.20 10.30 9.20 11.30 10.15 1,20 p.m. 12.5 2.50 12.15 345 1.30 4.0 2-35 445 245 5-37 4.30 7.0 5-35 8.25 What is the arithmetic average time of the journey? What is the difference between it and the actual time which most trains take in doing the journey? Which time do you regard as the more useful for the purpose of an average (using the term " average " in a general sense) ? State your reasons. 8. A concentric disc is removed from a circular disc 24 inches in diameter so as to leave only two-thirds of the original area. Find the radius of the disc removed. 9. For a certain book it costs a publisher ^100 to prepare the type and 2s. to print each copy. Find an expression for the total cost in pounds of x copies. REVISION EXERCISES 537 Also make a diagram on the scale of i inch to 1000 copies and i inch to jioo, to show the total cost of any number of copies up to 5000. Read off the cost of 2500 copies, and the number of copies costing ,525. 10. A very long strip of paper is rolled on a cylinder of wood i inch in diameter, and the total diameter of the roll is 2 inches. Find what would be the total diameter of the new roll if the same strip of paper were coiled on a cylinder of wood 2 inches in diameter. F 1. Employ an abbreviated method to divide 33-82289 by 66-4684 to six places of decimals (so that the error is less than 0-0000005). 2. Write down the nine numbers which immediately precede 8 1 and the nine which immediately follow, and show that their sum is exactly divisible by 9. Would the sum still be divisible by 9, if instead of a group of nine numbers on either side of 81, a group of five, six, or any other amount were, chosen ? Give your reasons. 3. Find the greatest number which will exactly divide both 867,502 and 1,000,369 ; find also the smallest number which is exactly divisible by both the numbers. 4. The surface of a sphere of radius r inches is 4?rr 2 square inches, where 77 = 3-14159. If TT is taken, instead, to be equal to 3j, find, roughly in square miles, the difference that this will make in the calculated area of the earth's surface, the earth being a sphere of radius 4000 miles. 5. How many moons would it take to form a body the size of the Earth ; the Moon's diameter being 2180 miles, and the Earth's diameter 7957-75 miles? 6. Genuine milk contains 88-75 P er cent, by bulk of water, 2-75 of fat and 8-5 of non-fatty solids. A purchaser buys 7 gallons of milk at ^d. a quart ; the milk on being analysed is found to contain 90-84 per cent, of water, 2-24 of fat, the residue being non-fatty solids. Find whether anything besides water has been added to it ; and find the sum of which the purchaser has been defrauded. 53 8 A MODERN ARITHMETIC 7. A right circular wooden cylinder has a length of 38 inches and a radius of i foot. At one end a hollow in the shape of a hemisphere, of radius i foot, is scooped out, and at the other end is scooped out a hollow in the shape of a cone, two feet high, and the base of which is one foot in radius ; find the volume of the remaining solid. Also find the weight of this solid, given that TT = 3-1416, that the density of the wood is nine-tenths of the density of water, and that a cubic foot of water weighs 1000 oz. 8. A person repays the principal of a debt by annual instalments, paying ^120 the first year, and each succeeding year 10 per cent, more than the year before : how much will he have repaid 10 years? lid in 3f the 9. A farmer bought 6 oxen and 100 sheep for ^336 ; of sheep, 4 died, and the rest were sold at ,2. js. 6d. each ; and 2 of the oxen fetched 1$ each. At what price must the remaining 4 oxen have been sold if the profit on the whole transaction amounted to 5 per cent. ? 10. From the table given below plot curves showing for the given period (i) the hour of sunrise, (2) the hour of sunset, (3) the duration of sunshine. Deduce (i) the latest sunrise, (2) the earliest sun- set, (3) the date of the shortest day. Sunrise. Sunset, h. m. h. m. Nov. 26 7 37 a.m. 3 57 p.m. Dec. 3 7 48 3 51 . ! 7 58 3 47 17 84,,. 3 47 2 4 88,, 3 50 3i 8 9 3 56 Jan. 787,, 44,, '4 8 4 4 14 21 7 56 4 26 REVISION EXERCISES 539 G 1. (i) A walk two yards broad taken off a square field all round it reduces its area by one quarter ; what was its area in square yards ? (ii) Is it possible for two numbers to have a 100 for their sum and 3000 for their product ? State the reasons for your answer. 2. A left .15,325 between B, C and D. To B he left such a sum that, after paying legacy duty at the rate of 10 per cent., B actually received ^600. C's legacy was to D's legacy as 3 to 2. What net sums did C and D receive if they paid legacy duty at the rate of 3 per cent.? 3. Find the square root of 87798284-89146121 ; and find the area of a face of a cube whose contents are 13634-789869 cubic feet. 4. Express 0-573 as a series involving geometrical progression, and thence show by summation that its value is ||f. 5. Describe any two methods by means of which a London merchant might settle a debt incurred with a merchant abroad. Find the cost of drawing at London a bill of 48,725 francs due Paris, at 25-22 cheque rate in 4 months. Bill Stamp ----- 0-5 per mille. Brokerage ----- i per mille. Rate of Discount in Paris - - 2| per cent. 6. Find, by using the tables, in what time a sum of money will treble itself at 5 per cent, compound interest. 7. Supposing the quantity of land under barley in England to be this year half as much again as that under wheat, and the quantity under oats to be equal to the other two together ; if the quantity under wheat next year be reduced by 25 per cent., and the quantity under barley increased by 5 per cent., the whole quantity remaining the same as before, by how much per cent, will the quantity under oats be increased ? 8. A right circular cylinder io| ft. long has a volume 528 cubic feet. Find the cost of gilding its convex surface at q\d. per square foot. (Use TJ =3}.) 540 A MODERN ARITHMETIC 9. The length of a room is to its breadth as 5 to 4. The flooring costs 46. 17 s. 6d. at js. 6d. per square yard, the walls cost ,30. 7s. 6d. to colour at ^\d. per square foot. Find the dimensions of the room, neglecting windows and door. HI. If 0-917 of a sovereign be pure gold, how much pure gold is there in 23-27 Ib. of standard gold? (Standard gold is the gold from which sovereigns are made.) 2. If i cub. ft. of brass weigh 551-25 Ib., how many cubic feet will there be in a block weighing 1290-95 Ib.? (State your answer to five significant figures.) 3. Express as and decimals of a ,, (a) ,173. $s. i%d., (b) ;i8. 17.?. 9!^. (True to the third decimal figure.) 4. Find the value of 547 f cwt. at i. 2.9. 8|</. per cwt. 5. Employ the given table to express the following lengths in English measure : (a) 1-24 metres, (b) 0-5321 metre ; state the answer true to three significant figures : 1 metre = 39-37 in. 2 =78-74 3 = 1 18-11 4 =157-48 5 =196-85 6. Coal is sold in England "free on board" a vessel for ijs. a ton, and in France the price realised is 30 francs per tonneau (1000 kilo.). What is total amount in francs per tonneau of freight and profit ? (i =25-23 francs.) 7. An estate consists of house property of gross rental of ,150, but on which the following payments are made : Insurance - - ^3 15 o Land Tax - - - 3 10 o Sewers Tax i 150 The losses on account of bad tenants, etc., amount to 12^ per cent, of the rental, ordinary repairs and expenses to 17^ per cent., and REVISION EXERCISES 541 collection of rent i\ per cent. Find the value of the freehold, calculated at 4 per cent, simple interest. 8. Which is the more advantageous of the following methods, which a London merchant might employ to settle in debt in Paris ? (i) Buying French 4 months Bills at 25-76. (ii) Buying Vienna Bills at 24-30, and selling these in Paris at 1-05. Brokerage i per mille. 9. Construct an isosceles triangle ABC, base BC 8 cm. and height AD 9 cm. Imagine a point O to start from D and to travel along the line DA to A (see Fig. 108). From measurements give a table of values for OA+OB+OC corresponding to values o, i, 2 ... 9 cm. of DO, and from these data draw a graph, representing DO by its actual length and OA + OB + OC by 5 times its actual length. Make the scales used and the position of the axes clear ; the zero may be allowed to fall beyond the paper. Determine from your graph how far O has travelled when the sum of its distances from A, B, C is the least possible. Show this position in your figure, and from measurement give the value of the angles AOB, BOC, COA. I 1. Find how many different integers, besides unity, divide 9009 without remainder. 2 . simplify V^' and calculate it to three places of decimals. 3. In 1875 there were 10,803,030 acres of cotton fields in the United States, and they produced 4,632,313 bales of cotton. In 1898 the acreage was 24,967,295, and the production 11,189,205 bales. Calculate, for each year, the average number of bales produced per 100 acres. (Give the nearest whole number.) 542 A MODERN ARITHMETIC 4. A person bought ^3000 worth of stock in 2\ per cents at 88. He sold out when these funds had fallen to 85, and invested the proceeds in the 6 per cent, railway preference shares at ^180 per ^ioo share. Find the alteration in his income, brokerage being neglected. 5. Find the rate of interest incurred by a person who buys the piano player below by monthly payments : " The piano player. ^29 cash, or 36 monthly payments of 2os." 6. Find the commercial discount at 3| per cent, on ^787. *3 S - $d- due six months hence. 7. A hollow spherical shell of metal has an internal diameter oi 10 inches, and the thickness of the metal is I inch : find the number of cubic inches of metal in the shell, and the cost of gilding the surface at $s. 6d. per square foot, taking TT=^. 8. A wine glass has its interior in the torm of an inverted cone of 2 inches diameter and 2 inches depth. Compare the interna' volume of such a glass with that of a tankard, the interior of whicr has the form of a frustum of a cone, the diameters of the ends being 3 inches and 4 inches respectively, and the depth 4-5 inches. (TT = 3j. 9. A commences business with a capital of ^4000, and aftei 4 months takes B into partnership with a capital of ^300. Tw< months later they take C into partnership with a capital of ^5000 At the end of the year their net profits amount to 16 per cent, on th( whole capital invested. What should each receive of the profits ? 10. Draw a plan of the field, and find its area in acres, roods perches from the following field-book entries in links : AD O 1 120 C 225 920 780 i8oE 520 60 F B 175 220 80 120 G o From A go East. REVISION EXERCISES 543 J 1. The following work is sent up by a person in answer to the question : "Find the L.c.M. of 165, 216, 209, 192, 200." 165, 216, 209, 192, 200 33, 216, 209, 192, 40 33,. 36, 209, 32, 40 L.c.M. = 5 x6x8xiix 3 6xi 9 x 5 33, 36, 20Q, 4, 5 =9,028,8oo. 5, 36, 19, \ 5 Show by an independent method of working the question that the answer is correct. The method being, nevertheless, fundamentally wrong, point out the errors, and explain why none of them affects the result. Also construct a simple example wherein an error of the same type does affect the answer, making it, e.g., three times as large as it should be. 2. Simplify ffi-j&-jh. 3. Express 0-0416 as a vulgar fraction in its lowest terms. Make use of this result in reducing 0-375416 to a fraction. 4. A workman is to be paid is. for his first day's work, is. id. for the second day, is. 2d. for the third day, and so on increasing at the rate of a penny each day that he works. Find how much more he earns in the second week than in the first, if he works six days in the week. 5. A closed cubical cistern, made of metal i centimetre thick, has an internal capacity of i cubic metre. Assuming the metal to be 7 times as heavy as water, find the weight of the cistern in kilograms. 6. How much stock at 97^ must I sell out in order to realise ,1300. And if, by investing this ^1300 in 4^ per cent, stock at 102, I increase my income by 8. 6s. 8d. per annum, what rate of interest was I receiving on the former stock ? 7. A hollow sphere of iron is of uniform thickness. Its external diameter is 9 inches and its weight is 72 Ib. Find the volume of the internal cavity. The iron weighs 480 Ib. per cubic foot, and the volume of a sphere of diameter d is $Trd z . If the diameter may be wrong by o-i inch, the weight of the 544 A MODERN ARITHMETIC sphere by an ounce, and the weight of a cubic foot of the iron by 10 lb., estimate the possible error in your result. 8. Find to the nearest penny the amount of ^453 in 3 years at 2| per cent, compound interest. 9. A retail bookseller is supplied by a wholesale firm, which allows discount off the published price of certain books (Class A) at 40 per cent. ; off all others (Class B) at 33^ per cent. At the end of a year he finds that he would have paid 2\ per cent, less than he actually did, had he accepted the tender of another wholesale firm, which offered a uniform discount of 37^ per cent. What percentage of his purchases (reckoned at published price) belonged to each class ? Give your answer to the nearest integer. 10. Draw, as accurately as possible, a fair-sized plan of a field from the following field-book reading. Then find the area in acres, roods and poles. The Field-Book. Links. To D 3250 2660 900 to C to E 400 2300 1980 300 to B to F 1600 1700 From A go North. K 1. Compute the value of correct to two places of decimals. Compute the sum of the infinite series of terms , + l + -L + _L_ + _' I 1.2 to six places of decimals. 2. The basin of the Severn is 4350 square miles in "area, an the yearly rainfall 29 inches. Half of the rain that falls is agai 1.2.3 1-2.3.4 REVISION EXERCISES 545 evaporated and the rest flows to the sea. Find, to the nearest million gallons, how much water flows to the sea per hour. (A cubic foot of water is 6-23 gallons.) 3. If the assessable value of the County of London is ^40,142,274, find what rate per pound must be levied in order to raise ,2,506,000. (Give your answer in current coin.) Also, by how much must the assessable value increase that an increase in the rate of id. in the pound should add ,250,000 to the amount raised ? 4. A solid iron cube, the edge of which is two feet in length, and a solid iron sphere, the radius of which is one foot, are thrown into a cubical tank, which is six feet across and is half filled with water. Find the rise of the surface of the water in inches, to five places of decimals, it being taken for granted that the cube and the sphere are both completely submerged. 5. Find the percentage error in the following approximations occasionally used in practical work : (i) Circumference of a circle is equal to three times the diameter added to one-seventh the diameter. (ii) To get the area of a circle : " Square the diameter, add the one-twentieth part of the result, and take three-quarters of the sum for the area." State your result true to two important figures, assuming that the true rules are : Circumference = TT (diameter). . Tr(diameter) 2 Area = - -*-, where ^ = 3-14159 true to the fifth decimal figure. 6. The breaking weight W in tons for a long iron column of length L feet, and diameter D inches, is sometimes calculated by the formula n 3 ' 76 W=i 4 -9^rr What is the diameter of a column 60 feet long, for which the breaking weight is 226 tons ? J.M.A. 2M 54 6 A MODERN ARITHMETIC 7. The difference between the simple and compound interest on a certain sum, for three years at four per cent, per annum, is ,12. 13-y. 4^. ; what is the sum? 8. There are four vessels of equal capacity ; the first is filled with spirit to an extent of |th, the second to th, the third to |th and the last to rd. The first is then filled up with water, and from this mixture the second is filled up, again from this second mixture the third is filled up, and in like manner the fourth from the third. What proportion of spirit to water is there in the fourth vessel ? 9. The population of a country increases regularly by i per cent, every year ; find, to three significant figures, what will be the total percentage of increase in 10 years. 10. The railway rate per ton for a certain class of goods is specified as follows : For any distance up to 20 miles, i penny per mile ; additional for the next 30 miles 0-8 penny per mile ; addi- tional for the next 50 miles 0-5 penny per mile ; additional for any further distance 0-4 penny per mile. There is also a fixed terminal charge of 3 shillings per ton irrespective of distance. Find the rate for 25, 75 and 145 miles, in pence per ton, and plot a curve showing the rate for all distances up to 200 miles. L 1. Some of the following statements are obviously wrong : indicate any errors that you can find, explaining your reasons. You need not correct the mistakes. The last figure in each answer is approximate only : (i) ( 4 .8i6)3=i 1 17. (ii) Vo-48i6=o-2i94. (iii) The area of a circle of radius 81-90 feet is 5268 square feet. 2. Work three fr the following (a, b, c, d) in the most intelligen way you can : (a) 30 miles an hour is how many feet per second ? REVISION EXERCISES 547 (b) If 15 per cent, of a certain amount is ^27. los. 6d., what would 1 6 per cent, of the same amount be? (c) 3-T42(7-86 2 -5-93 2 )- (d) (1-23456)2 to five significant figures. 3. Write down, or obtain expressions for the volume of (i) a pyramid on a square base ; (ii) a truncated pyramid on a square base. Find the volume of air contained in a street lamp with a flat top and bottom, the four slanting faces of the lamp being equal trapeziums with sides 6, 9^, 12 and 9^ inches long. 4. In the game ot dominoes a number of blocks are used. Each block is marked off into two parts, and on each of these parts a number of pips, between o and 9 inclusive, are marked. These are arranged in all possible ways, and the pips on the two parts of a block may be the same or different, and no two blocks are alike. How many blocks are there? nr~ l r~ n . 5. The formula is used by engineers. Find its value to two decimal places when #=1-05 and r=2. 6. A man invests ^1980 in the 3^ per cents at 99 and ^3220 in the 4^ per cents at 105 ; find the average rate of interest on his whole investment. 7. A cubic centimetre of an alloy of gold and silver weighs 13^ grams ; assuming that a cubic centimetre of gold weighs 19-6 grams and a cubic centimetre of silver 10-2 grams, and that the volume of the alloy is equal to the sum of the separate volumes of the gold and the silver contained in it, find the pro- portions by weight of gold and silver in the alloy. 8. Find the radius of a sphere, the volume of which is equal to the volume of a solid circular cylinder of height one foot and radius nine inches. 9. A merchant who sold his goods at a profit of 10 per cent, found that, when he allowed i\ per cent, discount off his selling price, his business increased by one-third. Find whether his 54 8 A MODERN ARITHMETIC total profits were increased or diminished by adopting this pi and in what proportion. 10. Make a list or table of the integral powers of 2 from index -2 to index +4. Use your values to draw a curve, the ordinates of which show the logarithms of numbers between 0-3 and 16-0 to base 2 ; use as unit for the numbers I cm. From your curve determine as accurately as you can log 2 i-7 and Iog 2 5'4- Use your curve to graduate the two pieces of gummed paper for use as a slide rule and fix the two pieces to a page of your book so as to show the product of 1-7 and 5-4. M 1. Express as a decimal : 2. A person writes down all the numbers from 21 to 40, both included, and proceeds to form the product of every two of them Without attempting to form the products, determine what per- centage of the whole number of them will be divisible by 6. 3. What is the saleable value per ton of the sweepings of z manufacturing goldsmith, in whose workshop the quantity o metal amounts to 0-02 per cent, of the sweepings ; assuming tha the metal is 10 carat gold, and that 30 per cent, of the value o the gold recovered is absorbed in the expenses of extraction, etc. Standard gold is 916-6 milliemes (p. 483) fine, and wortl ^3. 17-r. 6d. per oz. 4. There are four stations on a line of railway. How man; different kinds of single third-class tickets must be supplied fo use on the line ? 5. Three places, A, B, C, on a railway are passed by a trai: at 9, 9.40 and 10.32 ; another train passes A and B at 9.50 an< 10. 1 6, both trains travelling uniformly. At what time should th : second train pass C? If the first is travelling 24 miles an houi , what is the rate of the second ? [You may work by squared pape if you choose.] REVISION EXERCISES 549 6. A man invests ,7000 in the 4 per cents at 109! and has to pay an income tax of is. in the ; what amount of tax does he pay and what income has he left ? 7. An ordinary pail is 10 inches wide at the bottom and 14 inches at the top. If its height be 18 inches, find to the nearest tenth of a gallon how much water it will hold. [A cubic foot of water weighs 1000 oz., and a gallon of water weighs 10 lb.] 8. Almost all we know of Diophantus, the only algebraist Greece produced, is contained in the following epitaph from it find how long he lived : Diophantus passed of his life in child- hood, j^ in youth and } more as a bachelor ; five years after his marriage was born a son who died four years before his father at half his father's age at death. 9. The table below gives the relation between pressure and volume for i lb. of saturated steam, between certain limits of pressure. Plot a graph which will show this relation, and by counting squares on sectional paper, determine the area bounded by the curve, the horizontal axis or line of zero pressure, and the limiting ordinates (parallel to the line of zero volume). If for any small change of volume of the steam, the product of pressure in lb. per square foot and the change of volume in cubic feet represents the work done in foot lb., find how many foot lb. of work will be done in compressing the steam from a volume of 4-29 cubic feet to a volume of 1-53 cubic feet. Pressure in lb. per square inch. Volume in cubic feet. IOI-9 4-29 II5-1 3-82 129-8 3-42 145-8 3-07 163-3 2 76 I82-4 2-48 203-3 2 - 2 4 225-9 2-03 250-3 1-84 276-9 1-68 305-5 i-53 55 o A MODERN ARITHMETIC 10. ABCD is a quadrilateral such that AB =8 inches. BC =7 CD =9 DA =6 AC =7 Find the ratio of the two segments into which the diagonal BD is cut by the diagonal AC. N 1. Given that one foot = 30-48 centimetres, find, correct to the nearest foot, the side of a square field of area 15,600 square metres. 2. Find the weight in kilograms of an iron girder which is 5-4 metres long, 0-54 metre wide and 0-45 metre thick, having given that a cubic centimetre of iron weighs 7-76 grams. 3. An investor secures an annual income of j$$ from a capital of ,30,000 after paying income tax at $>d. in the pound. What sum should be invested in like securities to yield the same income after paying income tax at is. in the pound? 4. Define a sector. Two adjacent sides of a rectilineal field contain an angle of 67 30'. A cow is tethered at the angular point. Find to the nearest foot the length of the tether so that the cow may graze on 3 acres of land. 5. The following is an advertisement from a daily paper : " New and well built house for sale, price ,265 cash down, or 2. 8j. monthly for 15 years. To what rate per cent, simple interest is this equivalent ? 6. A bicycle wheel 28 inches in diameter makes 132 revolutions per minute. Express its speed in miles per hour. (7r = 2 T a .) A cyclist with a 28-inch wheel says that he can find his speed in miles per hour by simply counting the number of " clicks " made by his cyclometer in 5 seconds, each click indicating a revolution of the wheel. Test this assertion. 7. The annual average depth of rainfall for the three years 1879, 1880, 1 88 1 at a certain place was 24-98 inches ; the succeeding REVISION EXERCISES 551 three years it was 29-62. The year 1883 was the rainiest, when there fell 4-8 inches more than in 1882, 6-36 inches more than in 1884, and 7-47 more than in 1880. The year 1881 was short of the preceding year by only 0-17 inch. Find the depth of rain that fell in each of the six years. 8. A fundholder directs his broker to purchase eight ,100 shares in a certain mine, quoted at 272^ per share. To accomplish this he authorises him to sell out ^850 stock in the 3 per cents at 95|, and ^1300 stock in the 4^ at 105^. The broker's charge on each of the three transactions is |th per cent. What had the broker to receive upon the whole ? 9. A person C buys an estate, and some time afterwards disposes of it to another D for ^5400. At this price it yields D 2f per cent. C previously obtained 3 per cent, for his invest- ment, but his income, owing to less careful management, was ^36 less than D's. What did C originally pay for it ? 10. Show by a graph (unit 2-5 cm.) how the sum of a number n and its reciprocal - varies as the number changes from 0-2 to 6-5 ; use the graph for solving the equation n + -=a, (i) to find n when = 4-8, (ii) to find a when 72 = 3-24. 1. What is the difference between the compound interest on ^100 for two years, according as the interest is paid yearly or half- yearly, 4 per cent. ? 2. The legal weight of a 2o-yen piece is 257-21 grains, the sterling value ^2. os. iifdl, the weight of an English sovereign being 123-27 grains. What is the fineness of the 2o-yen piece? 3. A person buys ^6000 Argentine 5 per cent, stock when at 93. On the price rising to 95, he sells out, reinvesting the proceeds in Chinese 6 per cents. He finds that his new income is 3 more than his old. What was the price of the Chinese 6 per cents? Brokerage ^ per cent, on the Argentines and 5 per cent, on the Chinese. 55 2 A MODERN ARITHMETIC 4. The salary of a clerk begins at ,50 a year, and every year he receives a rise of .5. What is his salary in his 3oth year of service, and what has he received altogether at the end of the 29th year ? 5. A gasometer is in the form of a cylinder with a rounded top, and the following dimensions are given : diameter = 28 feet, height at edges 14 feet, height in the middle 16 feet. Draw a section of the gasometer on the scale of i inch to 10 feet, and calculate the number of cubic feet of gas that the gasometer will hold. Take the volume of the portion at the top (above 14 feet) as half the volume of a cylinder of the same base and height, and the area oi a circle as 3-14 times the square on its radius. 6. I know that the time of the half-swing of a pendulum / feet long is either 2ir\ seconds or TT\ seconds, where \32-2 \32-2 77=3-1416 ; I am uncertain which of these is correct, but I know that the half-swing of a pendulum a yard long takes about a second. Find from these data, with as little work as possible, which is the correct expression. 7. A platinum bar expands TTirroo f ' ts length at o C. for a rise of temperature of i C., and copper expands j-z^ for the same Find the lengths at o C. of two bars, one of platinum and the other of copper, such that, whatever be the temperature, the differ- ence of their lengths may be always 50 centimetres. 8. The populations of two towns are 107509 and 189160 their birth-rates per thousand are 27-9 and 25-7. Find to the same degree of exactness the birth-rate for the two towns taken together. 9. Evaluate 2^(/ 8 -/ 1 ) log^-log^' where TT= 3-142, k= 0-74, *i = 69-4, '2 = 82-3, r\= 1-25, r^= 1-55. 10. A person changes English money into French at the rate 01 25 francs for ji, and then changes the money back into English at the rate of 9^. per franc. Does he gain or lose, and how much does the difference amount to on ^100? REVISION EXERCISES 553 P 1. In 1901 the population of the United Kingdom was 41,800,000, and 34,500,000 cwt. of sugar were imported. How many pounds (to the nearest whole number) did we use per head in that year ? 2. Each member of a society subscribes as many pence as there are members in the society ; the total subscription is ,29. 8s. : how many members are there ? 3. Express 0-7962 as a vulgar fraction in its lowest terms, without assuming any rule in the process. Find the sum of all numbers between one and a million that are divisible by 119. 4. A refiner has 10 Ib. of gold 20 carats fine, and melts it with 1 6 Ib. of gold 18 carats fine : how much pure gold must he put into it to make the whole 22 carats fine? [24 carats fine is pure gold, 20 carats is 20 parts gold and 4 parts alloy.] 5. The sides of a rectangle are measured as a inches and b inches respectively, and its area calculated. If the true lengths are within x per cent, of these measurements, express in sq. inches the greatest possible error in the calculated area. Give this error to the nearest looth of a sq. inch when <2=9'7, ^ = 8-6, x= i. 6. Give an expression for the number of bricks measuring 8 by 4 by 2 inches that it takes to build a wall a yards long, b feet high and c inches thick. Give the result to the nearest thousand when ^ = 70, 6 = 6, c=i6. (Space occupied by mortar is to be disregarded.) 7. A and B would take 15 days to do a piece of work. They work together for 5 days and then call in C, and the three finish in 6 days more. C does in 2 days what A does in 3. Find their times separately for doing the work and thus compare them as workmen. 8. Equal sums are invested in 3 per cent, stock at 80, 4 per cent, at 90 and 5 per cent, at 100. The price of each stock rises ^10, so the ist and 3rd are sold and the proceeds invested in the 2nd. If the rise in income be \. 10^., find the whole sum invested. (No brokerage is to be reckoned.) 554 A MODERN ARITHMETIC 9. Electric trams start from a terminus every 4 minutes when in motion run at 10 miles an hour. Stations are ^ mile apart and there is a stop of a minute at each. Find the greatest and least distance apart of two successive trains. You may treat the question by arithmetic or graphic methods. 10. A length of 10 feet of a water pipe, f-inch in diameter, freezes. In freezing any volume of water expands to 1-087 times that volume. Sometimes this freezing bursts the pipe, sometimes the ice occupies an additional length of pipe. What additional length of pipe is necessary in the present case? If no additional length of pipe is available, to what diameter does the water expand in freezing? EXAMINATION PAPERS 555 TYPICAL EXAMINATION PAPERS. PART I. Cambridge Local (Preliminary). I. 1. Divide one hundred and two million three hundred and twelve thousand seven hundred and eighty-three by eighty-seven, and express your answer in words. 2. Add together ^73 8 9- 1 s - nfc^, ^36- 13-?- 8^., ,94. 19^. 4d.. ^2765. I4J. cfed., i8j. 7^-, ^784- 4-r- i<*/., and ,3065. gs. *]\d. 3. Find the cost of 796 pints of spirit at \is. jd. per gallon. 4. If 3 cwts. i qr. ii Ibs. cost ,1. 5^., what would one ton cost at the same rate ? 5. Find the greatest common measure of 42889 and 56903. 6. Reduce to their simplest forms (0 'rs-^+^o; (2) (i&xeio-Knftxffl. 7. Divide 0-566 by 0-0807 to four places of decimals. 8. Express -891 of ,1 in shillings and pence correctly to the nearest penny. 9. Find, correct to the nearest centime, the cost of 287 metres 7 centimetres at 5 francs 70 centimes per metre. 10. Find the simple interest on ,985. los. for 150 days at 3 per cent, per annum. 11. By selling a horse for ,57. 12s. a dealer gains 6| per cent. What was the original price of the horse ? 12. An estate is let to three tenants, of whom the first occupies two-fifths of the whole, the second three-eighths, and the third the remainder, which contains 25 acres 2 roods 24 poles. What is the size of the whole estate ? Cambridge Local (Preliminary}. II. 1. Divide three hundred and fifty million three hundred and one thousand three hundred and two by seven hundred and nine. 556 A MODERN ARITHMETIC 2. Add together 479. i8j. 8^., ^7036. 9* i<**, ^87. 12*. *]\d., ^3874. 7-r. i \\d., igs. 5^., ^4287- 5*. 6</., ^39- 14* 5K and ^ : 76- i&. 3. A man spends on the average 4*. <&d. per day. How much "does he spend in a year of 365 days ? 4. A plot of building land containing 3 acres 3 roods 25 poles is sold for ^1312. los. What is the price per acre ? 5. Reduce to their simplest forms (i) i$Vxjtf$; .(2) (|-f+fH(i+f+*> 6. Simplify, without reducing to vulgar fractions, (0-437 x 4-8) +(0-056 x 0-4). 7. Express 0-7187 of a mile in yards correctly to the nearest yard. 8. A merchant buys 235 metres 50 centimetres of silk for 659 francs 25 centimes. Find, correct to the nearest centime, the price per metre. . 9. Find the simple interest on ^375 for i\ years at i\ per cent. per annum. 10. A grocer mixes 28 Ibs. of tea costing is. 4%d. per Ib. with 1 1 Ibs. costing is. id. per Ib. Find the cost of the mixture per Ib. Find also, correct to one place of decimals, what is the grocer's gain per cent, if he sells the mixture at is. \o^d. per Ib. 11. Two bicyclists, A and B, start at the same instant to ride from Cambridge to London, a distance of 52^ miles, and travel at uniform speeds of \i\ and iif miles per hour respectively. How far will B be behind when A has reached London ? Cambridge Local (Preliminary}. HI. 1. Divide sixty millions eight hundred and forty-two thousands five hundred and thirty-five, by five hundred and seven ; and express the answer in words. 2. Add together 5*. 7^., 2. 6s. t&d., ^3041. js. iij^/., 253. 4s. gd., 476. i2s. 5<tf, ^219. 15^. iod., and 4006. 6s. 8|^. EXAMINATION PAPERS 557 3. The workmen employed in a certain factory earn on the average $s. i\d. each per day, and 234 men are employed. Find the total sum paid in wages for a week of six days. 4. Find the greatest common measure of 39483 and 138993. 5. Find the amount of tax on ^5354. 13^. 4^/., at the rate of is. J\d. in the pound. 6. Simplify (i) ifx3t-5-H; (2) (3rir-i&)-K3* + ii). 7. Divide 2789-44 by 0-0368, and 2-78944 by 368. 8. Find the interest for three months on ^720 at the rate of 3f per cent, per annum. 9. Express 0-5347 of an acre correctly to the nearest square yard. Find it also correctly to the nearest square foot. 10. Given that a metre contains 39-37 inches, express five miles in kilometres and metres, correct to the nearest metre. 11. If oranges can be bought at the rate of three for twopence, how many must be sold for a shilling to gain eighty per cent, profit on the cost ? Cambridge Local (Junior). A. 1. Find the highest common factor of 1881, 1976, and 4332. 2. How many seconds are' there between 2 p.m. on Monday, December 10, and 4.30 p.m. on the following Saturday ? 3. Simplify Divide 230-1052 by 0-00137. 4. Add together 4-15 of 2. 15.?. and 1-4 of half-a-crown. 5. Find the cost of 21 cwts. 2 qrs. 10 Ibs. of cocoa at ^3. js. %d. per cwt 6. Gold wire, ot a certain thickness, is worth i franc 56 centimes per. centimetre : what would be the cost of 2 metres 9 centimetres of this wire ? 7. Find the simple interest on ^1049. js. 6d. for 30 days at 4^ per cent, per annum. 55 8 A MODERN ARITHMETIC 1. A cubic foot of water weighs 1000 oz., and a gallon of water weighs 10 Ibs. Find the number of cubic inches in a pint. 2. A contractor engaged to construct 4 miles of railway in 184 days ; but 77 men working for 52 days made only one mile. How many additional men had he to employ, supposing all the men worked at the same rate as before, in order to complete his contract at the time specified ? 3. In a cricket match between two schools A and B, A made 85 more runs than B in the first innings ; but in the second innings B made 64 per cent, more runs than A, and won the match by 43 runs. What were the scores in the second innings ? 4. A hare makes 9 leaps in the same time as a dog makes 4, but the dog's leap is 7 feet while the hare's is only 3 feet ; how many leaps will the dog have to make before catching the hare, supposing the hare to have a start of 16 yards on a straight road ? 5. A merchant in New York buys icoo metres of silk in Brussels at 13 francs 54 centimes the metre : what is the cost, to the nearest dollar, assuming that 25 francs 20 centimes are worth 4-839 dollars ? 6. A person invests ,299 in consols ; on the price of ,100 consols being increased by ^3. los. he sells out, making a profit of jn. 7s. 6d. At what price did he buy the consols ? Oxford Local (Preliminary}. I. 1. Divide ,735 1. i6s. 6%d. by 71. 2. Find the Lowest Common Multiple of 48, 66, 27, and 72. 3. Simplify 3^ x 4^ + 2&. 4. Multiply 34-725 by 0-0128. 5. Express |- as a decimal. 6. Express 35-. ^\d. as a fraction o 7. Find the rent of a farm 25 acres 3 roods 16 perches in extent at \. \6s. 8d. an acre. EXAMINATION PAPERS 559 8. If 5 cwt. 3 qrs. 20 Ib. of tea cost ,44. $s. 4^., what is the value of 2 cwt. 16 Ib. ? 9. At what rate per cent, per annum, simple interest, will ;66. 1 3-y. Afd. amount to ,68. 1 5 s. in i year 3 months ? Oxford Local (Preliminary). II. 1. Multiply 28. iu. *]\d. by 79. 2. Find the Greatest Common Measure of 5434 and 8436. 3. Add together 2 J and 4^. Also multiply 2| by 1 5f . 4. Find the value of ^ of 7. $s. i\d. 5. Divide 15-2559 by 50-6. 6. Reduce 3^. qd. to the decimal of ,5. 7. Find the cost of 6 tons 7 cwt. 21 Ib. at 2. 6s. %d. per ton. 8. If 4 acres 3 roods 14 perches of land cost ,516, what should 9 acres i rood 20 perches cost ? 9. Find, correct to the nearest farthing, the simple interest on ^650. 4s. gd. for 5 years at 4^ per cent, per annum. Oxford Local (Preliminary}. III. 1. It ^1028. 175-. 4\d. is divided equally among 67 people, how much will each receive ? 2. Find all the common measures of 364 and 546. 3. Simplify (2H4)-K2f-il-)- 4. What fraction of 2. $s. 6d. is 2s. 2d. ? 5. Multiply 23-625 by 0-0034, and prove your answer by division. 6. Express 0-8125 as a vulgar fraction. 7. What is the cost of 6 cwt. 3 qrs. 20 Ib. at two guineas per cwt. ? 8. A field of 4 acres 2 roods 16 poles is rented at ,13. \6s. ; what is the rent of a similar field of 5 acres 3 roods 12 poles ? 9. A housekeeper takes three half-pints of milk each week-day and one pint on Sunday, and her bill for the week comes to is. What is the price of milk per quart ? 560 A MODERN ARITHMETIC Oxford Local (Preliminary). HIGHER ARITHMETIC. 1. Find the square root 01 0-00034969. 2. Linoleum is laid on a floor whose dimensions are 18 feet by 14 feet, so as to leave everywhere a margin of 2^ feet. What will the linoleum cost at 2s. 6d. per square yard ? 3. In what time will ^175 amount to ^197. i$s. at 4 per cent per annum simple interest ? 4. What is the cost of 157 centimetres of a material sold at 2 marks 30 pfennigs per decimetre ? 5. A man leaves ,6400 to be divided among 4 sons, 3 daughters, and 3 nephews. If each daughter receives three times as much as each nephew, and each son five times as much as each nephew, how much does each nephew receive ? 6. A man holding ^3000 of a 3 per cent, stock sells out when the stock is standing at 95, and invests the proceeds in a 5 per cent, stock at 1 14. What is the change in his income ? 7. A man travels from A to B to buy goods which he can get 10 per cent, cheaper in B than in A. If the expenses of the journey are 15 shillings and he makes a clear saving of ten shillings, what does he pay for the goods ? Oxford Local (Junior). 1. Simplify ( 2 i + 4 i- ii)-^^- 2 l +I i.). 2. What fraction of ^i. os. 6%d. is 4.?. lod. ? 3. Find (correct to the nearest penny) the rent of a field of 1 6 acres i rood 34 poles at ^5. i?s. 6d. an acre. 4. Show that to four places of decimals *$ is a correct approximation to the square root of 5. 5. If 25 francs = 1, and a gram =0-002204 lb. avoirdupois, find (in French money) the price of a kilogram of an article which costs a shilling an ounce. EXAMINATION PAPERS 561 6. A builder undertakes to complete a contract in 40 days. He employs 1 5 men, and at the end of 26 days only half the work is done. What is the least number of extra men that he must, engage, in order to complete the work in the specified time ? 7. Find (correct to the nearest penny) the compound interest on ,1500 for 3 years at 3! per cent, per annum. 8. The internal dimensions of a box without a lid are as follows : length 4 feet 2 inches, breadth 2 feet 4 inches, depth 2 feet i inch. Find, correct to the nearest penny, the cost of lining it with a material costing 6d. per square foot. 9. A bankrupt's liabilities are ^1350. His assets are nominally ,970, part of which consists of debts owing to him. He only succeeds in recovering 60 per cent, of his debts, and pays his creditors 12 shillings in the pound. What is the amount of the debts originally owing to him ? 10. Which is the better investment: a 5 per cent, stock at 132 or 3^ per cent, stock at 94? Give reasons for your answer. College of Preceptors (Third Class). L 1. (i) Write down the number three hundred and forty-one million three hundred and eighty thousand three hundred and fifty- four. (ii) Divide 9876543 by 567 (using factors), and explain how you find the remainder. (iii) Express the quotient in words, 2. A train is made up of an engine and nine carriages. If the weight of the engine be 36 tons 2 cwt. 3 qr. 21 lb., and that of each of the carriages 11 tons 15 cwt. 3 qr. 19 lb., find the whole weight of the train. 3. A grass field contains 10 acres 2 roods 36 poles. If 162936 square feet be ploughed, find how many acres, roods, square yds., etc., remain under grass. 4. A landlord's net rental, after paying income tax at the rate of is. in the ;, is ^7101. 14^. 6d. Find his gross rental. J.M.A. 2N 5 6 2 A MODERN ARITHMETIC 5. Add together 3f, 4$, 5$, 6|, 7^ 5 and subtract |f from 6. Simplify, 7. What do you mean by a decimal ? What do the 3 and 6 respectively mean in 38-046 ? Express as vulgar fractions in their lowest terms : 0-02, 0018, 2-25, 1-005. 8. Multiply 36-9 by 0-0058 ; and divide 1-392 by 0-00116. 9. If 3 tons 1 8 cwt. of lead cost ^100. 5$. 3^., what is the value of 3 cwt. 2 qr. of the same ? 10. Find by Practice the rent of 69 acres 2 roods 28 poles at 2. 5-r. per acre. 11. At an election where only seven-eighths of the whole number of voters voted, one of the two candidates received a fourth as many votes again as the other, and beat him by 896 votes. What was the whole number of voters ? College of Preceptors (Professional Preliminary]. 1. A light was seen at intervals of 13 sec. It was seen the first time at i hr. 54 min. 50 sec. A.M., and the last time at 3 hr. 17 min. 49 sec. A.M. How many times was the light seen ? 2. If 257 sheep and 37 oxen were bought for ^1187. 9^., and the average price of an ox was ,18. Js. 6d., what was the average price of a sheep ? 3. Explain how to obtain quantities of sand weighing 2 lb., 13 lb., 23 lb., and 39 lb. respectively from a large heap of sand, using scales sufficiently large to weigh 7 lb. of sand at a time, and only two weights, one of 4 lb., the other of 7 lb. 4. Simplify {5H+3H+2dfr}-H8$- 3 -2}. 5. Find the value of 0-6037 + 1% +0-023-3^, in decimal form, correct to the nearest figure in the fifth decimal place. 6. Express the sum of 2 tons 2 lb. and \\ of 6 tons 5 cwt 3 qr. 9 lb. as a decimal of 25 tons. EXAMINATION PAPERS 563 7. What is the rent, to the nearest centime, of 27-3 ares of land, at the rate of 5830 francs for 89-8 hectares ? [i franc = 100 centimes.] 8. Calculate, to the nearest penny, the sum to be paid to a creditor whose debt is ,183. 13.5-. lod. when his debtor pays I4s. J\d. in the . 9. The base of a rectangular tank is horizontal, and the area of the base inside is 37 square metres. What is the depth of the water, to the nearest centimetre, when the tank contains 17 kilolitres of water? [A litre is a cubic decimetre.] 10. A square field whose area is 128634 sq. yd. is to be enclosed with barbed wire placed at heights of i, 2, 3, and 4 feet above the ground. What length of wire, to the nearest yard, will be wanted if the length of wire required for each circuit is 3 per cent, greater than the perimeter of the field ? 11. Eleven persons contributed a certain sum. Nine of them gave ^5 apiece, and the other two persons gave $ and 10 more respectively than the average subscription of the eleven subscribers. What sums did the two persons give? 12. If 46 per cent, of a regiment containing 1150 men are recruits, how many recruits must be taken out in order that 25 per. cent of the remainder may be recruits? College of Preceptors {Second Class or Junior). /. 1. How much will a million and a quarter of oranges cost at 4^. a dozen ? 2. How many sheep worth ,3. 17 s. 6d. each must I give in exchange for 7 oxen worth ^27. 2s. 6d. each ? 3. I design a room 18 ft. square and n ft. high. Its windows, doors, and fireplace will take up 10 sq. yds. How many pieces (each 12 yds. long) of paper, i\ ft. wide, will give me what I require for the walls ; and how many yards of carpet, 22 in. wide, shall I need for the floor, allowing a space 7 ft. by i ft. for the hearthstone ? 5 6 4 A MODERN ARITHMETIC 4. My lad has gone to the post, distant 121 yds. less than half a mile away. If he walks down to it at the rate of 3 miles an hour, and returns up from it at the rate of 2 miles an hour, find how long, to the nearest second, he will be gone. 5. Simplify 6. Add together 40, 0-04, 0-0004, 4000-4 ; and divide the result by 400. 7. Obtain the square root of 1787-5984 5 and square I54'35- 8. What will the Simple Interest on ^14,5 amount to in 3^ years at 3^ per cent. ? 9. If I give ,10,500 for 741 acres of land, how much do I pay per are? [i hectare = 2-47 acres.] 10. Write down 3^ cub. millimetres as a decimal of a cub. decimetre ; and i\ square centimetres as a decimal of a square metre. 11. The "Victorious" left Malta on Saturday, May I3th, 1905, at 2 p.m., steaming 10 knots an hour. The Royal Yacht, with the Queen on board, caught her up and steamed past her at the rate of 1 5 knots an hour, at 8 p.m. on the Monday following. At what time and date did the latter leave Malta; and at what distance from Malta were the two ships abreast, their speeds being uniform throughout ? College of Preceptors (Third Class). II. 1. The average daily supply of water within the area of the Metropolitan Water Board during the month of January last was two hundred and four millions nine hundred and ninety-four thousand gallons. The estimated population within the same area being six millions seven hundred and seventy-four thousand eight hundred and thirty-nine, calculate to the nearest gallon the supply per head per day. EXAMINATION PAPERS 565 2. A and B set out walking from the same place in opposite directions at the rate of 3^ miles an hour and 3| miles an hour respectively. In what time will they be 58 miles apart ? 3. A sum of .25. 14.?. $d. is divided among 3 men and 5 women so that each man may have twice as much as a woman. What sum will each man receive? 4. Find, by Practice, the value of 159 cwt. 3 qr. 22 Ib. at 2. \2.s. 6d. per cwt. 5. What fraction added to the sum of i| + 4f + 7 will make a sum total of 15? 7. Reduce 0-0375 and 0-0025 to vulgar fractions in their lowest terms ; and find the value of 0-018 of 4. $s. $d. 8. Multiply 0-59324 by 0-00675 5 and divide 5-084976 by 16-86. 9. If 2 acres I rood 27 poles of land cost ^258, what should 1 8 acres 3 roods of the same sort of land cost? 10. The debts of a bankrupt amount to ^4269. is. od. He has assets amounting to ^2846. os. %d. How much in the pound can he pay his creditors ? 11. Calculate in tons the weight of water in a reservoir I acre in area, and having a depth of 18 inches, a cubic yard of water weighing 15 cwt. College of Preceptors (Second Class or Junior). II. 1. Find in millimetres, correct to the nearest millimetre, the circumference of a carriage wheel which makes 59,173 turns in travelling 170 kilometres. 2. Express in pounds, shillings, pence, and farthings the least exact sum of money greater than 1541 half-crowns, and the greatest exact sum of money less than 1957 florins. How many different exact sums of money are there which are both greater than 1541 half-crowns and less than 1957 florins? 566 A MODERN ARITHMETIC 3. What is the greatest common measure of 12,936 and 20,328 ; also what numbers between 20 and 50 will exactly divide both of them? (2M-fr)of9*-i| 2+&rf9J 5. Obtain, in decimal form, the values of O-3XO-2+O-2I xo-4 and 0-4 x 0-2- 0-2 1 x 0-3 ; and divide the sum of the squares of the two values by 14-5. What is the square root of that sum ? 6. Determine the difference, correct to the nearest second, between 29-5306 days and 27-3217 days. 7. If 457 quarters of wheat cost ,109. 4.?. 8d. more than 659 quarters of oats at i6s. ^d. per quarter, what was the price of i quarter of wheat ? 8. Find, by Practice, correct to the nearest penny, the value of 13 cwt. 3 qr. 20 Ib. at ^59 per ton. 9. Assuming that A spends ,30,000 in 365 days, and that B spends 24^. in 7 days, express B's daily expenditure, (i) as a fraction, and (ii) as a decimal, of A's average daily expenditure in the latter case, correct to the nearest figure in the sixth decimal place. 10. A dealer bought some pears at qs. per hundred and sold them for 2. $s. \od. If he obtained an average price of is. 4^. per dozen, what whole profit, and what rate per cent, of profit, did he make ? 11. If a sheet of lead 10-9 metres long and 19 decimetres wide weighs 1747 kilogrammes, and i cubic centimetre of lead weighs 1 1-4 grammes, what is the thickness of the sheet, correct to the nearest tenth of a millimetre ? College of Preceptors (Third Class). III. 1. Divide eight million one hundred and fifty-three thousand and seventy-eight by sixty-three, using factors, by short division ; and give the method by which you find the remainder. EXAMINATION PAPERS 567 2. ,840. 9^. 3^/. is to be divided equally among 65 persons. How much does each receive ? 3. A plot of land 45 ft. by 16 ft. produces 32 stones of potatoes. What is the produce per acre ? 4. Find, by Practice, the cost of 11 tons 12 cwt. 2 qrs. 14 Ib. of goods at ,5. 6s. %d. per ton. 5. What fraction must be subtracted from the sum of to leave 18 exactly? 6. Simplify (3f + 4& of if H 7. What is meant by the local value of a digit ? Write down the local value of each figure in the number 78-543. Express of half-a-crown as a fraction of a guinea. 8. Multiply 0-00806 by 7-5, and divide 17-175 by 0-0125. 9. A train travels 30 miles an hour. Express this speed in feet per second. 10. A ship worth ,20,000 belonging to four persons, A, B, C, and D, became a total loss. A owned , B , C &, D J. She was insured for ,15,000. What will each receive, and what will he lose? 11. A brick measures 9 in. long, 4-5 in. broad, and 2-75 in. deep. How many such bricks are contained in a wall 16-5 ft. long, 13-5 in. broad, and 4-5 ft. high ? Pupil Teacher Candidates (Junior Grade). 1. A man borrows .300 from one person at 3 per cent, and also ,750 from another person at 4^ per cent. At what rate does the man pay interest on the whole sum of money borrowed ? 2. How many lengths of wire 0-85 inch long can be cut from a piece 37-076 inches long ? What is the length of the remainder ? 3. A cistern open at the top is 13 ft. long, 8 ft. broad, and 5| ft. deep. What will it cost to paint the inside at ^\d. per square yard ? 4. A man after spending three-quarters of his money and two- fifths of the remainder had 8^. 9^. left. What did he spend ? 568 A MODERN ARITHMETIC 5. A grocer adds to a chest of tea worth 2s. 6d. a lb., twenty per cent, of its weight of another kind of tea worth is. <)d. a lb. What is the mixture worth ? Pupil Teacher Candidates {Senior Grade). (Questions I to 5 were not arithmetical questions?) 6. The heights of the barometer at certain times on March 1 6th are given in the following table : Time of observation 2 p.m. 4.15 8.0 10.30 Midnight Height of barometer\ in inches - -J 29-4 29-3 29-1 28-95 29-05 Construct on squared paper a graph showing this variation. 7. Three trucks contain 25 tons 16 cwt. of coal. The first truck contains 0-411 and the second 0-214 of the whole. What weight of coal is there in the third ? 8. A can mow a field in 4^ days, B in 5^ days. If they work together for two days, what fraction of the field will be left unmown? 9. How many times can a cup containing \ of a pint be filled from a can containing 2 galls, i quart of milk ? 10. A rectangular room measuring 16 ft. by 14 ft. 6 inches is carpeted so as to leave a width of i ft. 3 inches of bare board on every side. What is the area of the carpet ? 11. A shopkeeper sells an article so as to make 25 per cent, en the cost price. If he had sold it for sixpence more he would have made thirty per cent. What is the selling price ? 12. A man went to market with 8^. gd. in his pocket. He sold 45 sheep, and then bought 3 cows for ^18. 5* each. He then had 25. 6s. <)d. What sum did he get for the sheep ? EXAMINATION PAPERS 569 Preliminary Examination for the Certificate. PART I. 1. Annie was born on August 3rd, 1903, while her baby brother was 106 days old on last Christmas Day. How old was Annie in years and days when the baby was born ? 2. I have 12 reels of cotton each containing a single piece of cotton 220 yards in length. Explain clearly how many pieces of cotton can be cut off, so that each may be 75 feet in length and without a join. 3. Continue the decimal fraction 7-3 by writing in the second decimal place a number that means one-sixth as much as the 3, and give reasons for your answer. 4. What is the nearest sum to T 2 T of^i2 that can be paid in our coinage (a) if farthings are allowable, (b) without the use of farthings ? 5. Write out so as to be understandable by a child acquainted with multiplication, division, and fractions but not with ratio, the solution to the following sum : If it cost ^3 to carry a weight for 144 miles, what will it cost to carry the same weight for 117 miles at the same rate per mile? 6. A florist buys flowers at the rate of is. for 14 ; he sells them at is. ^d. a dozen. How many must he sell to make a profit of 1 1 s. qd. ? 1. A brick measures 9 inches, by 4^ inches, by 3 inches. How many will be required for a solid wall 21 feet 9 inches long, 13^ inches thick, and 4 feet 6 inches high, when the thickness of the mortar is neglected ? 8. Assuming that the product of 376038 and 294876 is 110884581288, find the product of 376038 and 294879^. 9. Write out fully in words the work involved in dividing 4731 by 3, being careful to state fully the true value of every dividend, quotient, and remainder. 570 A MODERN ARITHMETIC 10. A person who has 0-3 of a mine, sells f of his share for ^1275. What is the value of all the shares in the mine? 11. State clearly why the suggested answers to the following questions are obviously wrong : (1) Multiply 9f} by 6f. Answer 69. (2) How much tea at is. jd. a Ib. must be mixed with a cwt. of tea at 2s. 6d. a Ib. in order that the mixture may be worth is. i \\d. a Ib. ? Answer 49 Ib. (3) Multiply 573 by 963. Answer 550799. (4) Which is the greater 26 cubic feet or a cubic metre ? Answer 26 cubic feet. 12. A coal merchant bought 44 tons of coal at an average price of i2s. 6d. a ton. When doing this, he gave 2os. a ton for 9 tons. 9_y. 6d. a ton for 13 tons, and us. jd. a ton for 12 tons. What was the price per ton that he gave for the remainder ? 13. A man borrowed ,20 and agreed to pay is. a month for the use of the money. At what rate per cent, did he pay interest ? 14. A grocer weighed out what he thought were i Ib. packet.' of tea. He then discovered that his Ib. weight was really 15! oz How much tea was there in 100 packets, and how many complete packets did he make out of i cwt. ? 15. Work as clearly as possible, by the shortest method you cat think of, the following : (a) I993^-I993- (b) Find the cost of 2400 articles at 19^. ii^d. each. (d) The cost of 8250 bricks at los. 6d. a thousand. TYPICAL EXAMINATION PAPERS PART II Civil Service Commission. 1. A sample of petroleum weighed about 864 ounces per cub. foot. Find how many grams it weighed per litre, (i kilo = 2-206 lb., i metre = 3-28 ft.) Show that for any substance the number of ounces per cubic foot is nearly equal to the number of grams per litre. 2. Fig. i is drawn on a scale of 10 feet to the inch, and represents the end view and side view of a shed, the roof and side walls of which are made of corrugated iron. Find the cost of the roof and side walls at \\d. per square foot. FIG. i. 3. The following statistics are given in the Census of 1901 District. Number of Acres. Population. London Westmoreland England 74,839 505,330 32,611,033 4,536,541 64,409 30,829,695 Find, in each district, the population per acre correct to three decimal places, that is, to the nearest thousandth. And calculate to two significant figures the ratio of the density of London to that of England and the ratio of the density of Westmoreland to that of England. 4. The pressure of the atmosphere is 15 pounds per square inch. When a bicycle tyre, of capacity A cubic inches, is inflated by a pump of capacity B cubic inches, the pressure of air in the tyre, after n strokes of the pump, exceeds the atmospheric pressure by 572 A MODERN ARITHMETIC wxB-i-A times the atmospheric pressure. In a certain case the pump is 10 inches long and |-inch in diameter ; the tyre may be treated as a cylinder 88 inches long and if inches in diameter ; and the capacity of a cylinder a inches long and b inches in diameter is 0-79 xaxl>xl> cubic inches. Find the capacity of the tyre and of the pump, and the pressure of the air in the tyre after 40 strokes, each to two significant figures. 5. A book of gold leaf contains 25 leaves, 3^ inches square and jssWff f an mc h thick. If a cubic inch of gold weighs 0-697 lb., and i lb. of pure gold is worth ^61. 19^., find the value of the gold in the book to the nearest penny. 6. Fig. 2 shows on a scale of i to 2000 a level field ABCD, con- taining a pond E, whose average depth is 60 cm. below the level FIG. 2. of the field. It is proposed to remove the water from E and fill Uj with earth taken from the surface of the rest of the field. Find by pricking the diagram through on to your book and counting squares, what fraction of the whole field is covered by the pond and hence calculate how many centimetres the surface of the fiel< will have to be lowered in order to fill up the pond. If the work costs 2s. per cubic metre moved, what is the tota cost? TYPICAL EXAMINATION PAPERS 573 Civil Service Commission. HIGHER ARITHMETIC. 1. A man buys a house for .1150. There is a ground rent of a year, and annual repairs come, on an average, to i^ per cent. on the purchase price of the house. At what rent must he let the house to clear 8 per cent, per annum on the purchase price ? 2. By pricking off Fig. 3 (shown only half-size) on to your book, and counting squares, find its area in square centimetres. Hence FIG. 3. find the weight of steel rail, of this section, for a single line railway 17 kilometres long. Answer in tonnes, (i cub. centimetre of steel weighs 7-8 grams.) 3. A man takes a boat and rows up stream. When he has gone \\ miles up stream he passes a bottle floating down with the stream. He goes on up the river for another 18 minutes and then turns. When he gets back to the starting- place he finds that the 574 A MODERN ARITHMETIC bottle has just arrived there. Assuming that his speed through the water and the speed of the stream are both uniform, find the speed of the stream in miles per hour. 4. A wine merchant buys in Bordeaux a cask of claret, and the freight to London is 1. It contains 208 litres of wine at 75 centimes the litre, and for the cask itself he has to pay 12 francs. He has to pay duty is. $d. per gallon, and bottles the wine at a cost of 2s. a dozen bottles. If he sells it at iSs. a dozen, what percentage of this price is profit ? (i = 25 francs = 2500 centimes ; 6 bottles = I gallon =4-546 Ijtres.) 5. From the following table find (to the nearest shilling) the value per cwt. of coffee in 1903, in 1904, and in 1905. Find also the percentage falling off in 1905 in quantity and in total value as compared with 1904, each percentage to the nearest integer and reckoned on the 1904 numbers : Quantity Imported. Cvvts. - 1,143,526 1903 1904 1905 6. In Fig. 4 ABC re- presents the circumference of a new grindstone weighing 102 Ibs. Find its weight when it is worn down to DEF. Ignore the hole through the stone. 1,055,866 956,226 Value in . 3,134,924 3.329,598 2,707,169 TYPICAL EXAMINATION PAPERS 575 7. If a water-pipe is L yards long, d inches in diameter, and one end is H feet higher than the other, then *J($df xTJT L gallons of water will flow through the pipe in a minute. Use this formula to find how many gallons per minute will flow through a pipe a mile long, 4| inches in diameter, one end being 38 feet higher than the other. 8. Two crews, A and B, row a race. For the first 4 minutes A row 33 strokes a minute and their boat travels 36 feet through the water at each stroke. B, who have been rowing 32 strokes a minute, are then 80 ft. behind. A then drop to 31 strokes a minute, but their boat moves 37 ft. through the water at each stroke, while B quicken to 33 strokes a minute, but their boat moves at each stroke i| ft. less than it did before. If they go on like that, how long will it be before the boats are level, and how far will they then have rowed ? Board of Education (Certificate Examination). (Questions 7, 9, 10, 11, 12 were Algebraical.) 1. Evaluate as shortly as you can the three most important figures in the product of 4680-13576 and 286423-5791, stating shortly the reasons for your procedure. 2. How -would you teach children who are acquainted with the mensuration of a circle to find the volume of a cylinder ? Three cylinders are of the same height and of the same material. Prove that a triangle whose sides are equal to the diameters of the cylinders will be right-angled if one cylinder weighs as much as the other two together. 3. Describe a method of practically demonstrating to children the operations performed in the sum usually set down as follows : . s. d. 3)i5 . 16 . 2 5)5. 5.4 and 2 over. i . i . o and 4 over. Explain the method by which we can calculate the value of the remainder. 57 6 A MODERN ARITHMETIC 4. Two trains are travelling in opposite directions at uniform speeds. They each pass a telegraph post in the same time, and they pass each other in five seconds. If the lengths of the trains added together be 264 yards, and the faster train's speed exceed that of the slower by 25 %, find the length and speed of each train. 5. A rectangular board measuring 6 decimetres by half a metre is just large enough to display 120 circular medals, all of the same size, arranged edge to edge in 12 rows. Find (a) The area of the board that is left uncovered by the medals ; () The area of a face of one medal. [Take ^=3-1416.] 6. Simplify without using unnecessarily lengthy methods (a) ( 4 -68) 3 + 3X 1-32 x( 4 .68) 2 + 3 x(i -32)2 x 4-68+ (i^) 3 . 13 (c) 2117x1883-1113x887. 8. Two trains, moving at the rates of 45 and 47^ miles an hour pass simultaneously but in opposite directions through a statior at 9 a.m. Draw a graph to show their distance apart at any time up to noon. Use the graph to find their distance apart at 10.4* a.m., and the time at which they are 200 miles apart. College of Preceptors (First Class}. MENSURATION. 1. Write down the tables (a) for lineal measure, (b) for super ficial measure. 2. The base of the largest of the Egyptian Pyramids is a square A side of this square is 693 feet. What extent of ground in acres roods, and perches is covered by this Pyramid ? TYPICAL EXAMINATION PAPERS 577 3. A rectangular court is 120 feet long and 90 feet broad, and has a path of 10 feet round it. Find the cost of covering the path with flag-stones at 4^. 6d. per square yard and turfing the court itself at gs. 6d. per 100 square feet. 4. The area of a triangular field is 7 acres 3 roods 14 poles. The length of its base is 16^ chains. Find the length of its per- pendicular in chains from the vertex to the base. 5. The sides of a triangular field measure : AB, 1400 ; BC, 1760 ; CA, 3000 links. It is let at 2 per acre. Find the annual rent. 6. The cost of fencing a circular piece of land was ^59. 1 1 s. %d. at 2s. id. per yard. Find the length of a straight path running from side to side through the centre. 7. The circumference of a wheel is 2 metres 7 centimetres. Find the number of revolutions it will make in travelling 2 kilo- metres 33 decametres 91 decimetres. 8. The diameter of a circular well is 9 feet 4 inches, and the depth of the well is 30 feet. Find the cost of sinking it at the rate of 13^. 6d. per cubic yard. 9. Give the definition of a prism, also of a prismoid, with figures ; and find the solidity in cubic inches of a prismoid whose greater end is 18 and its breadth 8 inches, the length of the less, end being 12 and its breadth 6 inches, the perpendicular height being 18 inches. 10. Work done in digging, etc., is estimated (English) by the number of pounds weight raised i foot, or (French) by the number of kilogrammes raised I metre, called "metre-kilogrammes." Express a metre-kilogramme in foot-pounds. [Given I metre =39'37 inches, i kilogramme =15430 grains, i foot =12 inches, i pound =7000 grains.] J.M.A. 2 o 57 8 A MODERN ARITHMETIC 11. A field is in the form of an irregular pentagon, required from the particulars given below (a) To draw a diagram of the field on one sheet of paper. (b) To draw up the Field-Book on one sheet. (c) To find the area of the field ABODE. A horizontal line AD is first measured = 351 links. At 71 links along AD from A a perpendicular measuring 84 links falls to B. At 205 links along AD a perpendicular is drawn on the opposite side of the horizontal line, measuring 129 links. At 305 links along AD a perpendicular is drawn to it, ending in C on the same side as the first perpendicular, measuring 84 links. Cambridge Local (Junior}. MENSURATION AND SURVEYING. 1. A building is in the form of a cylinder 40 ft. in diameter and 25 ft. high, surmounted by a hemispherical dome of equal diameter. Estimate the cost of painting its whole external surface, at \o\d. per square yard. 2. A rectangular cottage measures 28 ft. 6 in. by 24 ft. o in. The rain which falls on its roof is led in equal shares into three equal water-butts, whose depth is 3 ft. 4 in., and average diameter 2 ft. 6 in. Find how many inches of rainfall will fill the butts. 3. Draw on the scale of one chain to the inch the outline of an irregular pond about half an acre in extent, and explain briefly how you would make a chain survey to determine its actual area. TYPICAL EXAMINATION PAPERS 579 4. On the scale of two chains to the inch draw a plan of the roughly triangular field surveyed according to the following field book : OC o 1311 |^Line 3 58 1 100 103 900 157 700 672 T A 738 1 88 500 T 11 S 200 Tie line. T to A. OB o O * Line i Line 2. B to C. A o 856 >* Line i 33 700 B o 1108 xr Line 2 551 84 IOOO 500 22 116 800 400 37 98 600 245 56 200 28 80 A o O * Line 3 Co / Line 2 Line i. A to B Line 3. C to A. bearing N.N.W. FIG. 5. Diagram of field. 5 8o A MODERN ARITHMETIC 5. Find the area of the property whose plan, on the scale of 4 chains to i inch, is here given. FIG. 6. The London Chamber of Commerce. 1. The weight of a rectangular block of iron is 865-432 grams, to the nearest milligram, the weight of each cubic centimetre being 7-734 grams, also true to the nearest milligram. On being measured with an instrument reading to the nearest hundredth of a centi- metre, the length is found to be 7-63 cm., and the breadth 5-48 cm. Find the thickness of the slab, as accurately as the data allows. 2. The following is an advertisement taken from a weekly paper : "Handsome Cuckoo Clock offered at 12s. 6d. post free for full cash with order, or post free on receipt of 5^. and your promise to pay 2s. 6d. monthly until 15^. is paid." If money be reckoned as earning simple interest, what rate per cent, per annum of interest is the above equivalent to ? 3. A steam engine employed in pumping water from a coal mine is found to consume an amount of coal proportional to the product of the amount in gallons of water pumped up, and the depth in feet from which the water is raised. If 112 Ibs. of coal is burnt in pumping 9600 gallons of water from a depth of 600 feet, how much coal will be burnt when 12,000 gallons are raised from a depth of 7 50 feet? TYPICAL EXAMINATION PAPERS 4. Prove the truth of the following rule for expressing sums of money in pounds and decimals of a pound, " Count each shilling as 0-05 of a ,, express the pence as farthings, increasing the number by one twenty-fourth of itself, regard each modified farthing as o-ooi ,." What would be the corresponding rule for converting decimalized sums into s. d. within a farthing? (a} Express ,35. 6s. ^\d. as a decimal of a . (ft) Express ,0-8732 in shillings and pence within a farthing. (c) Find the value of 7856 articles at 6. 2s. M. each. 5. A person instructs a stockbroker to buy 3000 shares at o. 17^. 6d. per share, the settlement with his broker is deferred however for 7 months, at the end of which time the price of the shares having risen to i. 2s. gd., the person instructs his broker to sell. If the broker charges 8 per cent, per annum for the money value of the shares, what payment is due to or from the broker ? (brokerage $d. per share on each transaction). 6. Find the average temperature at Barnstaple for the twelve months stated, also the average temperature at Oxford during July, 1903 : Barnstaple. Oxford. 1903. July .... 60-4 ? August 58-6 58-7 September - 567 56-8 October 53-4 52-2 November - 47-3 44-1 December - 41-4 38-6 1904. January 42-4 39-4 February - 41-0 39-i March 42-3 41-0 April - May - --- 48-9 52-3 49'3l 52-7 June - - - - 57-4 57-2 Average for 12 months, ? 49-2 If in each case the data be known to the nearest tenth of a degree, state the maximum possible error in each answer. 582 A MODERN ARITHMETIC 7. Average limestone consists of 112 parts of carbonate of lime, to 9 of clay and 29 of sand. Find its percentage composition. If on burning, 56 per cent, of the carbonate yields lime, what quantity of limestone is required to yield I ton 8 cwt. of lime ? 8. An iron rod is known to weigh 17-346 lb. per foot run. What would be the corresponding figuies in grams per centimetre ? State clearly the assumptions you make. 9. What sum of money must be invested at 4 per cent, per annum compound interest (interest payable yearly), so as to gain 37- 5-f- & as interest in three years ? Oxford Local (Senior). 1. Resolve 462 into prime factors, and hence find all the factors of 462 which are greater than 50. 2. Use decimal fractions to find the value of 1.2.3 1-2.3.4 1.2.3.4.5 1.2.3.4,5.6 3. If i metre = 3-2809 feet, find, correct to the nearest foot, the difference between 161 kilometres and 100 miles. 4. Find how many cubical blocks, each of whose edges is 62-5 centimetres long, could be made out of 75 cubic metres of clay : and find, also, to the nearest cubic centimetre, how many cubic centimetres of clay would remain over. 5. (i) How many decimal places must generally be retained in working a sum in compound interest in which money is expressec as the decimal of a , in order that the result may be correc to the nearest penny ? (2) Find to the nearest penny the compound interest or 4387. 12s. 6d. for 3 years at 4 per cent, per annum. 6. A slow train leaves A at 1.30 p.m. and reaches B, no mile away, at 7 p.m., while an express leaving B at 4 p.m. reaches / at 6 p.m. Assuming that each train travels uniformly, find th time at which the trains will meet. TYPICAL EXAMINATION PAPERS 583 7. A tea-dealer allows 10 per cent, discount off the marked price of his tea. On tea marked is. 8d. per Ib. he makes a profit of T2rr per cent. The duty on tea is reduced one penny per Ib., and the dealer then marks his tea at is. id. per Ib. (discount being allowed at the same rate as before). Find his percentage of profit now. 8. A chemist has two bottles containing mixtures of nitric acid and water. The liquid in the first bottle contains 25 per cent. of acid, and the liquid in the second bottle contains 37-5 per cent, of acid. How much liquid must be taken from each bottle in order to obtain a third mixture containing 5 litres of acid and 9 litres of water ? 9. Two brothers receive equal money legacies. One of them invests all his money in India 3! % Stock at 103^, The other invests part of his money in 2^ % Consols at 86| and the rest of it in Railway 4% Preference Stock at 112^. If the two brothers receive equal incomes from their investments, find the ratio of the money invested in Consols to the money invested in the Railway Stock. Cambridge Local (Senior). 1. Find the cost of 8 tons 14 cwt. 3 qr. 12 Ib. at ,3. 15^. lod. per ton. 2. Find the Greatest Common Measure of 78913 and 501866. 3. Simplify 4. Find to the nearest foot the value of 0-04 of 5-745 miles ; and reduce 9. 14^. ^d. to a decimal of 10 guineas. 5. . If 29 metres 75 centimetres of cloth cost 190 francs 40 centimes, what is the cost of 22 metres 5 centimetres ? 6. A square field has an area of 5^ acres. Find to the nearest penny the cost of a fence round it at 3^. gd. per yard. 584 A MODERN ARITHMETIC 1600 7. Find to the nearest penny the compound interest on ,1600 for 3 years at 3 per cent, per annum. 8. A grocer mixes 60 Ibs. of tea, which cost is. <)d. a pound, with 80 Ibs. of tea, which cost 2s. ^d. a pound, and sells the mixture at 2s. jd. a pound. What is his profit per cent, on his outlay ? 9. Find the quarterly income derived from the investment of ,4256 in a 3f per cent, stock at 95. 10. The railway line between two towns A and B is 55^ miles long. The 4.42 p.m. express train from A arrives at B at 6.3 p.m. ; and the 4.30 p.m. express train from B arrives at A at 5.42 p.m. Assuming that each train travels at a uniform rate, find the time at which they meet one another and the distances of the meeting place from the two towns. TABLES 585 TABLES. DESCRIPTION OF STANDARDS OF WEIGHT AND MEASURE. (Based on a Table exhibited in the Victoria and Albert Museum, Soiith Kensington.} BRITISH STANDARDS. The yard of 36 inches is the unit or standard measure of extension, from which all other imperial measures of extension, whether linear, superficial, or solid, are ascertained. The pound is the unit of the measure of weight from which all other imperial weights, or measures having reference to weight, are ascertained. The one seven thousandth part of the pound is a grain. One sixteenth part (437^ grains) of the imperial standard pound is an ounce avoirdupois. The ounce troy consists of 480 grains and is now the only legal standard troy weight. All articles sold by weight must be sold by avoirdupois weight ; except that (1) Gold and silver, and articles made thereof, including gold and silver thread, lace or fringe, also platinum, diamonds, and other precious metals or stones, may be sold by the ounce troy or by any decimal parts of such ounce and (2) Drugs, when sold by retail, may be sold by apothecaries' weight. [The "apothecaries' " ounce is of the same weight as the troy ounce of 480 imperial grains.] The unit or standard measure of capacity from which all other measures of capacity, as well for liquids as for dry goods, are derived, is the gallon containing ten imperial standard pounds weight of distilled water (weighed in air against brass weights) at the temperature of 62 Fahrenheit and with the barometer at 30 inches. METRIC STANDARDS. The third Schedule to the Weights and Measures Act, 1878, sets forth the equivalents of imperial weights and measures and of the weights and measures therein expressed in terms of the metric system, and Sect. 21 of the Act provides that a contract shall not be invalid if weights or measures 5 86 A MODERN ARITHMETIC of the metric system, or decimal subdivisions of imperial weights and measures, are used in it. The unit of Length is the Metre = 39- 3708... inches. ,, ,, Surface ,, Are =100 sq. metres = 119-6033 sq. yds. ,, Capacity,, Litre =ywo" cub. metre= 1 76077 pints. ,, ,, Weight Gram =15-4323487 grains. (A gram is the weight of a cubic centimetre of distilled water at 4 centigrade. ) BRITISH TABLES OF WEIGHT AND MEASURE. LINEAR MEASURE. 12 inches (in.) = i foot (ft.). 3 feet = i yard (yd. ). 1760 yards =i mile (mi.). GUNTER'S CHAIN MEASURE (LINEAR). 100 links or 4 poles = I chain=22 yards. 10 chains = I furlong. 8 furlongs = I mile. MEASURES USED FOR SPECIAL PURPOSES. I Geographical mile or Nautical mile=i-i5 ordinary or Statute mile. League =3 miles. Fathom -6 feet. Pace (Military) = 2 feet 6 inches. Pace (Geometrical) = 5 feet. Ell = i i yards. Cubit =18 inches. Quarter or Span =9 inches. Hand Palm Nail 12 Lines = =4 inches. =3 inches. =2-25 inches. =i inch. points SQUARE MEASURE. I2 2 = 144 square inches (sq. in.) = i square foot (sq. ft.). 3 2 = 9 square feet 10,000 sq. links = i sq. chain. 10 sq. chains = i acre. 640 acres = I sq. mile. 10,000 sq. links = 484 sq. yards = 1 6 = i square yard (sq. yd.). 40 sq. poles = i rood. 4 roods = i acre. poles = i sq. chain. TABLES 587 CUBIC MEASURE. I2 a = 1728 cubic inches (cub. in.)= i cubic foot (cub. ft.). 3 3 = 27 cubic feet = i cubic yard (cub. yd.). MEASURES OF CAPACITY. 4 gills =i pint (pt.). 2 pints = i quart (qt.). 4 quarts = i gallon (gal.). 2 gallons = i peck (pk. ). 4 pecks (8 gal.)=i bushel (bus.). 8 bushels =i quarter (qr.). MEASURES USED FOR SPECIAL PURPOSES. 4| gallons = I pin. 9 gallons = I firkin. 36 gallons = i barrel. 54 gallons = I hogshead. I hogshead of Ale = 54 gallons. I hogshead of Claret = 46 gallons. I hogshead of Port =57 gallons. APOTHECARIES' MEASURE. 60 minims = i fluid drachm (f 3). 8 drachms = I fluid ounce (f g). 20 ounces = i pint (pt.). 8 pints (160 f)= i gallon (gal.). AVOIRDUPOIS WEIGHT. (The grain has the same weight in each of the three tables beiow, and is the seven-thousandth part of the Standard Pound.) 16 drams (dr.) = i ounce (oz.). 16 ounces (7000 grains) = i pound (lb.). 14 pounds =1 stone (st). 28 pounds (2 st-) = i quarter (qr.). 4 quarters (i!2 Ibs.) = I hundredweight (cwt.). 20 hundredweights (2240 lbs.)=i ton (t.). TROY WEIGHT. 24 grains (gr.) = i pennyweight (dwt.;. 20 pennyweights (480 grains) = i ounce (oz.). 12 ounces (5760 grains) = i pound (lb.). A MODERN ARITHMETIC APOTHECARIES' WEIGHT. 20 grains (gr.) =i scruple O). 3 scruples = I drachm (3). 8 drachms (480 grains) = I ounce (). 12 ounces (5760 grains) = I pound (lb.). METRIC TABLES OF WEIGHT AND MEASURE. LINEAR MEASURE. 10 millimetres (mm.)=i centimetre (cm.). 10 centimetres 10 decimetres IO metres 10 decametres 10 hectometres = 1 decimetre (dm.). = i metre (m.). = I decametre. = I hectometre. = i kilometre (km.). SQUARE MEASURE. IO 2 = 100 square millimetres = IO 2 = 100 square centimetres = IO 2 = 100 square decimetres = io 2 loo square metres io 2 = 100 square decametres = square centimetre. square decimetre. square metre centiare. square decametre are. square hectometre hectare. square kilometre. principally in dealing IO 2 = IOO square hectometres = The terms centiare, are and hectare are used with land measures. CUBIC MEASURE. io 3 = 1000 cubic millimetres = i cubic centimetre (c.c.). IO 3 = 1000 cubic centimetres = I cubic decimetre. io 3 =iooo cubic decimetres = i cubic metre = i stere. MEASURES OF CAPACITY. IO centilitres = I decilitre. IO decilitres = i cubic decimetre = I litre. IO litres = I decalitre. IO decalitres = I hectolitre. io hectolitres = I kilolitre =i cubic metre. TABLES 589 WEIGHT. 10 milligrams = 10 centigrams = IO decigrams = 10 grams = 10 decagrams = 10 hectograms = centigram. decigram. gram(gr.). decagram. hectogram. kilogram (kilo.). (The gram is the weight of I cubic centimetre of water at the tempera- ture of 4 C. ) SUMMARY OF METRIC MEASURES. Fraction or Multiple of Standard Measure Length. Volume. Weight. O-OOI O-OI Millimetre Centimetre Millilitre Centilitre Milligram Centigram O-I Decimetre Decilitre Decigram I Metre Litre (1000 c.c.) Gram 10 Decametre Decalitre Decagram IOO Hectometre Hectolitre Hectogram IOOO Kilometre Kilolitre Kilogram APPROXIMATE EQUIVALENTS OF IMPERIAL WEIGHTS AND MEASURES IN TERMS OF METRIC UNITS. IMPERIAL TO METRIC. Linear Measure : I inch i foot (12 inches) 1 yard (3 feet) - Square Measure : I square inch - - - I square foot (144 square inches) I square yard (9 square feet) Cubic 'Measure : I cubic inch .... I cubic foot (1728 cubic inches) I cubic yard (27 cubic feet) = 2-54 centimetres. :O-3O metre. 0-914 metre. :6-45 so i- centimetres. = 929 sq. centimetres. : 0-836 sq. metre. ; 1 6-387 cub. centimetres. ; 0-028 cub. metre. = 0-764 cub. metre 590 A MODERN ARITHMETIC Measures of Capacity : i gill i pint (4 gills) - I quart (2 pints) 1 gallon (4 quarts) Apothecaries' Measure : I minim - - - '.' i fluid scruple I fluid drachm (60 minims) I fluid ounce (8 drachms) - I pint 1 gallon (8 pints or 160 fluid ounces) Avoirdupois Weight : i grain i dram I ounce (16 drams) - 1 pound (16 ounces or 7000 grains) - Troy Weight-. I grain ....... I pennyweight (24 grains) - I troy ounce (20 pennyweights) - Apothecaries^ Weight : i grain I scruple (20 grains) - - - I drachm (3 scruples) - i ounce (8 drachms) - = 1-42 decilitres. : 0-568 litre. : 1-136 litres. = 4-546 litres. : 0-059 millilitre. : 1-184 rnillilitres. : 3-552 rnillilitres. = 2-841 centilitres. = 0-568 litre. : 4-546 litres. ; 0-065 gram. = 1-77 grams. ; 28-35 grams. : 0-4536 kilogram. =0-065 gram. = 1-55 grams. = 31-10 grams. = 0-065 gram. = 1-296 grams. = 3-888 grams. = 31-10 grams. APPROXIMATE EQUIVALENTS OF METRIC WEIGHTS AND MEASURES IN TERMS OF IMPERIAL UNITS. METRIC TO IMPERIAL. Linear Measure : I millimetre (mm.) (o-ooi m.) - - - =0-03937 inch. I centimetre (o-oi m.) - - - - =0-3937 inch. I decimetre (o- 1 m.) =3-937 inches. TABLES 591 1 metre (m.) - - I decametre (10 m.) i hectometre ( i oo m.) - i kilometre ( 1000 m.) - Square Measure : 1 square centimetre i square decimetre (100 square centimetres) i square metre (100 square decimetres) Cubic Measure : I cubic centimetre i cubic decimetre (c.d.) (looo cubic centi- metres) ------ I cubic metre (1000 cubic decimetres) Measure of Capacity. I centilitre (o-oi litre) .... I decilitre (o-i litre) 1 litre Weight : milligram (o-ooi grm.) .... centigram (o-oi grm.) - decigram (o-i grm.) gramme (i gm.) decagram (10 gm.) hectogram (100 gm.) .... 1 kilogram (1000 gms.) .... I gram (i gm.) - I gram(igm.) f 39-37 inches. = j 3-28 feet. 1 1-09 yards. = 10-936 yards. = 109-36 yards. = 0-62 mile. = 0-155 square inch. = 15-50 square inches. f 10-76 square feet. ~ 1 1-19 square yards. = 0-06 cubic inch. = 61-02 cubic inches, f 35-31 cubic feet. \i-3i cubic yards. = 0-070 gill. = 0-176 pint. = 1-76 pints. Avoirdupois. = 0-015 grain. =: 0-154 grain. = I> 543 grains. = I5-43 2 grains. = 5-644 drams. = 3-527 oz. _ f 2-20 lb. or ,:. \15432-3 grains. Troy. _ ^0-032 oz. troy. ~ 1 15-43 grains. Apothecaries. ("0-257 drachm. = 1 0-77 1 scruple. >*5*43 grains. 592 A MODERN ARITHMETIC MONEY TABLES. BRITISH. 4 farthings ({d. ) = 12 pence 2 shillings = 5 shillings = 20 shillings = 21 shillings = penny (d.). shilling (s.). florin. crown. sovereign or pound guinea. FRANCE AND OTHER COUNTRIES.- f centimes - -"| C franc (France, Switzerland, Belgium, etc.). 100 J centesnilos " I i J lire (Italy). j centimes - J peseta (Spain), lleptas - -J I drachma (Greece). I franc, lire, peseta, or drachma has a value of about cfed. GERMANY. IOO pfennigs = I mark. 3 marks = I thaler. 10 marks = I krone. I mark has a value of about is. UNITED STATES. 10 cents = i dime. 10 dimes (100 cents) = i dollar. 10 dollars = I eagle. I dollar has a value of about 4*. 2d. MEASUREMENT OF TIME. 60 seconds (sec. or ")= i minute (min. or ') 60 minutes = i hour (h. ). 24 hours =i day (d.). 7 days = i week (wk.). 28 days or 4 weeks = i lunar month (mon. ). 365 days = i ordinary or civil year (yr.). 366 days = i leap year. TABLES. SQUARE ROOTS AND RECIPROCALS. Numbers (n) from I to 100. n ^ I/fl n A I/* n *fc I/* 1 I -0000 I -0000 36 6-0000 0-02778 71 8-4621 0-01408 2 1-4142 o-5 37 6-0828 0-02703 72 ! 8-4853 0-01389 3 17321 Q-33333 38 6-1644 0-02632 73 8-5440 0-01370 4 2-OOOO 0-25 39 6-2450 0-02564 74 8-6023 0-01351 5 2-236I O-2 4O 6-3246 0-025 75 8-6603 0-01333 6 2-4495 0-16667 41 6-4031 0-02439 76 8-7178 0-01316 7 2-6458 0-14286 42 6-4807 0-02381 77 8-7750 0-01299 8 2-8284 O-I25 43 6-5574 0-02326 78 8-8318 0-01282 9 3-0000 O-IIIII 44 6-6332 0-02273 79 8-8882 0-01266 10 3-1623 0-1 45 6-7082 0-02222 8O 8-9443 0-0125 11 3-3166 0-09091 46 6-7823 O-O2I74 81 9-0000 0-01235 12 3-4641 0-08333 47 6-8557 0-02I28 82 9-0554 O-OI22O 13 3-6056 0-07692 48 6-9282 O-02O83 83 9-1104 O-OI205 14 37417 0-07143 49 7-0000 O-02O4I 84 9-1652 O-OII9O 15 3-8730 0-06667 50 7-0711 O-O2 85 9-2195 O-OII76 16 4-0000 0-0625 51 7-1414 0-OI96I 86 9-2736 0-OII63 17 4-1231 0-05882 52 7-2111 0-OI923 87 9-3274 O-OII49 18 4-2426 0-05556 53 7-2801 0-OI887 88 9-3808 O-OII36 19 4-3589 0-05263 54 7-3485 0-01852 89 9-4340 O-OII24 20 4-4721 0-05 55 7-4162 0-01818 90 9-4868 O-OIIII 21 4-5826 0-04762 56 7-4833 0-01786 91 9-5394 0-01099 22 4-6904 0-04545 57 7-5498 0-01754 92 9-59I7 0-01087 23 4-7958 0-04348 58 7-6158 0-01724 93 9-6437 0-01075 24 4-8990 0-04167 59 7-6811 0-01695 94 9-6954 0-01064 25 5-0000 0-04 60 7-746o 0-01667 95 9-7468 0-01053 26 5-0990 0-03846 61 7-8102 0-01639 96 9-7980 0-01042 27 5-1962 0-03704 62 7-8740 0-01613 97 9-8489 0-01031 28 5-29I5 0-03571 63 7-9373 0-01587 98 9-8995 O-OIO20 29 5-3852 0-03448 64 8-0000 0-015625 99 9-9499 O-OIOIO 30 5-4772 0-03333 65 8-0623 0-01538 100 IO-OOOO 0-01 31 5-5678 0-03226 66 8-1240 0-01515 32 5-6569 0-03125 67 8-1854 0-01493 33 5-7446 0-03030 68 8-2462 0-01471 34 5-8310 0-02941 69 8-3066 0-01449 35 5-9161 0-02857 70 8-3666 0-01429 N.B. The use of these tables can be extended by the methods suggested on pp. 310 and 320. J.M.A. 2P 593 LOGARITHMS.* 1 2 3 4 5 6 7 8 9 123 456 789 10 0000 0043 0086 0128 0170 4 913 17 21 26 30 34 38 0212 0253 0294 0334 0374 4 812 16 20 24 28 32 36 11 0414 0453 0492 0531 0569 4 812 15 19 23 27 31 35 0607 0645 0682 0719 0755 4 7 11 15 19 22 26 30 33 12 0792 0828 OSC4 0809 0934 711 14 18 21 25 28 32 0969 1004 1038 1072 1106 3 710 14 17 20 24 27 31 13 1139 1173 1206 1239 1271 1303 1335 1367 1399 1430 3 710 3 710 13 16 20 13 16 19 23 26 30 22 25 29 14 1461 1492 1523 1553 1584 369 12 15 19 22 25 28 1614 1644 1673 1703 1732 3 6 9 12 15 17 20 23 26 15 1761 1790 1818 1847 1S75 369 11 14 17 20 23 20 1903 1931 1959 19S7 2014 368 111417 192225 16 2041 2068 2095 2122 2148 358 111416 19 22 24 2175 2201 2227 2253 2279 358 10 13 16 IS 21 23 17 2304 2330 2355 2380 2405 358 101315 18 20 23 2430 2455 2480 2504 2529 257 101215 17 20 '22 13 2553 2577 2601 2625 2648 257 91214 161921 2672 2695 2718 2742 2765 257 91114 1618 21 19 278S 2810 2833 2856 2878 247 91113 16 1820 2900 2923 2945 2967 2989 246 81113 15 17 19 20 3010 3032 3054 3075 3096 3118 3139 3160 3181 3201 246 81113 15 17 19 21 3222 3243 3263 3284 3304 3324 3345 3365 3385 3404 246 81012 1416 IS 22 3424 3444 3464 3483 3502 3522 3541 3560 3579 3598 246 81012 141517 23 3617 3636 3655 3674 3692 3711 3729 3747 3766 3784 246 7 911 1315 17 24 3802 3820 3338 3856 3874 3892 3909 3927 3945 3962 245 7 911 12 14 16 25 3979 3997 4014 4031 4048 4065 4082 4099 4116 4133 235 7 910 12 14 15 26 4150 4166 41S3 4200 4216 4232 4249 4265 4281 4298 235 7 810 111315 27 4314 4330 4346 4362 4378 4393 4409 4425 4440 4456 235 689 111314 28 4472 4487 4502 4518 4533 4548 4564 4579 4594 4609 235 689 111214 29 4624 4639 4654 4669 4633 4698 4713 4728 4742 4757 134 679 101213 30 4771 4786 4800 4814 4829 4843 4857 4871 4886 4900 134 679 10 11 in 31 4914 4928 4942 4955 4969 4983 4997 5011 5024 5038 134 678 101112 32 5051 5065 5079 5092 5105 5119 5132 5145 5159 5172 134 578 91112 33 5185 5198 5211 5224 5237 5250 5263 5276 5289 5302 1 3 4 568 91012 34 5315 5328 5340 5353 5366 5378 5391 540S 5416 5428 134 568 91011 35 5441 5453 5465 5478 5490 5502 5514 5527 5539 5551 124 5 (5 7 91011 36 5563 5575 5587 5599 5011 5623 5635 5647 5G53 5670 2 4 567 8 10 lit 37 5582 5694 5705 5717 5729 5740 5752 57C3 5775 5786 2 3 567 8 910 33 5798 5S09 5321 5332 5343 5855 5866 5S77 5888 5899 2 3 567 8 910 39 5911 5922 5933 5944 5955 5966 5977 5988 5999 6010 2 3 457 8 916 40 6021 6031 6042 6053 6064 6075 6085 6096 6107 6117 2 3 5 6 8 910 41 6128 6138 6149 6160 6170 6180 6191 6201 6212 6222 2 3 5 6 789 42 6232 6243 6253 6263 6274 6284 6294 6304 6314 6325 2 3 5 6 789 43 6335 6345 o355 63G5 6>75 6385 6395 6405 6415 6425 2 3 5 6 789 44 6435 6444 6454 6464 6474 64S4 6493 6503 6513 6522 2 3 5 6 789 45 6532 6542 6551 65r,l 6571 6580 6590 6599 (5609 6618 2 3 5 6 780 46 6628 6637 6(546 6T.56 66(55 6675 66R4 6603 6702 6712 2 3 5 6 778 47 C721 6730 6739 6749 6758 6767 6776 6785 6794 6803 2 3 5 5 678 48 6S12 6S21 6S30 6839 6848 6857 6866 6875 6884 6893 2 3 4 5 678 49 6902 6911 6920 6928 6937 6946 6955 6964 6972 6981 123 4 5 678 Frorp Castle's Practical Mathematics for Beginners. (Macmillau.) 594 LOGARITHMS. 1 2 3 4 5 6 7 8 9 123 456 789 50 6990 6998 7007 7016 7024 7033 7042 7050 7059 7067 2 3 345 678 51 7076 7084 7093 7101 7110 7118 7126 7135 7143 7152 2 3 345 678 52 7160 7168 7177 7185 7193 7202 7210 7218 7226 7235 2 2 345 677 53 7243 7251 7259 7267 7275 7284 7292 7300 7308 7316 2 2 345 667 54 7324 7332 7340 7348 7356 7364 7372 7380 7388 7396 2 2 345 667 55 7404 7412 7419 7427 7435 7443 7451 7459 7466 7474 2 2 845 567 56 7482 7490 7497 7505 7513 7520 7528 7536 7543 7551 2 2 345 567 57 7559 7566 7574 7582 7589 7597 7604 7612 7619 7627 2 2 345 567 58 7C34 7642 7649 7657 7664 7672 7G79 7GS6 7694 7701 1 2 344 567 59 7709 7716 7723 7731 7738 7745 7752 7760 7767 7774 1 2 344 567 60 7782 7789 7796 7803 7810 7818 7825 7832 7839 7846 2 3 4 566 61 7853 7860 7868 7875 7882 7889 7C90 7903 7910 7917 3 4 566 62 7924 7931 7938 7945 7952 7959 79C6 7973 7980 7987 2 3 3 566 63 7993 8000 8007 8014 8021 8028 8035 8041 8048 8055 2 3 3 556 64 8062 8069 8075 8082 8089 8096 8102 8109 8116 8122 2 3 3 556 65 8129 8136 8142 8149 8156 8162 8169 8176 8182 8189 2 3 3 556 66 8195 8202 8209 8215 8222 8228 8235 8241 8248 8254 2 3 3 556 67 3261 8267 8274 8280 8287 8293 8299 8306 8312 8319 2 3 3 556 68 8325 8331 8333 8344 8351 8357 8363 8370 8376 8382 2 3 3 5 6 69 8388 8395 8401 8407 8414 8420 8426 8482 8439 8445 2 2 3 5 6 70 8451 8457 8463 8470 8476 8482 8488 8494 8500 8506 1 2 2 3 5 6 71 8513 8519 8525 8531 8537 8543 8549 cr r e 8561 85G7 1 2 2 3 455 72 8c73 8579 8585 8591 8597 8603 8609 8615 8621 8G27 1 2 2 3 5 5 73 8633 8639 8645 8651 8657 8663 8669 8G75 8681 8CS6 1 2 2 3 455 74 8692 8698 8704 8710 8716 8722 8727 8733 8739 8745 1 2 2 3 5 5 75 8751 8756 8762 8768 8774 8779 8785 8791 8797 8802 1 2 233 5 5 76 8308 8814 8S20 8825 8831 8S37 8S42 8S48 8S54 8859 1 2 233 5 5 77 8865 8871 8S76 8882 8887 8S93 8899 8904 8910 8915 1 2 233 4 5 78 8921 8927 8032 8938 8943 8949 8954 89GO 89C5 8971 1 2 233 4 5 79 8976 8982 8987 8993 8998 9004 9009 9015 9020 9025 1 2 233 445 80 9031 9036 9042 9047 9053 9058 9063 9069 9074 9079 112 233 445 81 9085 9090 9096 9101 9106 9112 9117 9122 9128 9133 112 233 445 82 9138 9143 9149 9154 9159 9165 9170 9175 9180 9186 112 233 445 83 9191 9196 9201 9206 9212 9217 9222 9227 9232 9238 112 233 445 84 9243 9248 9253 9258 9263 9269 9274 9279 9284 9289 112 233 445 85 9294 9299 9304 9309 9315 9320 9325 9330 9335 9340 112 233 445 86 9345 9350 9355 9360 9365 9370 9375 93SO 9385 9390 1 2 233 445 87 9395 9400 9405 9410 9415 9420 9425 9430 9435 9440 1 223 344 88 9445 9450 9455 9460 9465 9469 9474 9479 94S4 94S9 1 223 344 89 9494 9499 9504 9509 9513 9518 9523 9528 9533 9538 1 223 344 90 9542 9547 9552 9557 9562 9566 9571 9576 9581 9586 1 223 344 91 9590 9595 9600 9G05 9609 9G14 9619 9624 9628 9633 1 223 344 92 9638 9643 9647 9652 9657 9661 96G6 9671 9675 9680 1 223 344 93 9685 9689 9694 9699 9703 9708 9713 9717 9722 9727 1 223 344 94 95 9731 9777 9736 9782 9741 9786 9745 9791 9750 9795 9754 9800 9759 9805 9763 9809 9768 9814 9773 9818 Oil 223 344 96 97 9823 9808 9827 9872 9S32 9877 9836 9881 9841 9886 9845 9890 9850 9894 9854 9S99 9859 9903 9863 9908 Oil 223 344 98 9912 9917 9921 9926 9930 9934 9939 9943 9948 9952 Oil 223 344 99 9956 9961 9965 9969 9974 9978 9983 9987 9991 9996 Oil 223 334 595 TABLE OF DAYS. DAY OF THE MONTH. DAY OF THE YEAR. 1 JH % ! f 1 j>, "a ti 3 <J & & 8 i 1 1 I 32 60 91 121 IS 2 182 213 244 274 305 335 2 2 33 61 92 122 153 183 214 245 275 306 336 3 3 34 62 93 123 154 184 215 246 276 307 337 4 4 35 63 94 I2 4 155 185 216 247 277 308 338 5 5 36 64 95 125 I 5 6 1 86 217 248 278 309 339 6 6 37 65 96 126 157 187 218 249 279 310 340 7 7 38 66 97 127 158 1 88 219 250 280 3" 34i 8 8 39 67 98 128 159 189 22O 251 281 312 342 9 9 40 68 99 129 1 60 190 221 252 282 313 343 10 10 4i 69 IOO 130 161 191 222 253 283 3H 344 11 ii 42 70 IOI 131 162 192 223 254 284 315 345 12 12 43 7i 102 I 3 2 163 193 224 255 285 316 346 13 13 44 72 103 133 164 194 225 2 5 6 286 317 347 14 H 45 73 IO4 134 165 195 226 257 287 318 348 15 15 46 74 105 135 1 66 196 227 258 288 319 349 16 16 47 75 1 06 136 167 197 228 259 289 320 35 17 17 48 76 107 137 1 68 198 229 260 290 321 35i 18 18 49 77 108 138 169 199 2 3 26l 291 322 352 19 19 50 78 109 139 170 200 231 262 292 323 353 20 20 5i 79 no 140 171 20 1 232 263 293 324 354 21 21 52 80 III I 4 I 172 202 233 264 294 325 355 22 22 53 81 112 I 4 2 173 203 234 26 5 295 326 356 23 23 54 82 H3 143 U4 2O4 235 266 296 327 357 24 24 55 83 114 144 175 205 236 267 297 328 358 25 25 56 84 H5 145 176 206 237 268 298 329 359 26 26 57 85 116 146 177 207 238 26 9 299 330 360 27 2 7 58 86 117 147 178 208 239 270 300 331 36i 28 28 59 87 118 I 4 8 179 209 240 271 301 332 362 29 29 88 119 149 1 80 210 241 272 302 333 363 30 30 89 120 150 181 211 242 273 303 334 3 6 4 31 31 90 151 212 243 304 365 596 COMPOUND INTEREST TABLE (showing the sum to -which (a) i, (b) i yearly, will amount at Compound Interest). No. RATES PER CENT. OF YEARS. 2-5 3 3-5 4 4-5 5 1 (a) 1-02500 1-03000 1-03500 1-04000 1-04500 1-05000 (b) i- i- i- i- i- i- 2 (a) 1-05063 i -06090 1-07123 i -08 1 60 109203 1-10250 (4 2-025 2-03 2-035 2-04 2-045 2-05 3 (a) 1-07689 1-09273 1-10872 1-12486 1-14117 1-15763 (b) 3-07563 3-0909 3-10623 3-1216 3-I3703 3-I525 4 (a) 1-10381 1-12551 I-I4752 1-16986 1-19252 1-21551 (b) 4-15252 4-18363 4-21494 4-24646 4-27819 4-3ioi3 5 (a) 1-13141 1-15927 1-18769 1-21665 1-24618 1-27628 (b) 5-25633 5-309I4 5-36247 5-41632 5-4707I 5-52563 6 (a) 1-15969 1-19405 1-22926 1-26532 1-30226 1-34010 (b) 6-38774 6-46841 6-550I5 6-63298 6-71689 6-80191 7 (a) 1-18869 1-22987 1-27228 I-3I593 1-36086 1-40710 (b) 7-54743 7-66246 7-77941 7-89829 8-01915 8-14201 8 (a) 1-21840 1-26677 1-31681 1-36857 1-42210 1-47746 (b) 8-73612 8-89234 9-05169 9-21423 9-38001 9-54911 9 (a) 1-24886 I-30477 1-36290 1-42331 1-48610 I-55I33 (b) 9-95452 10-1591 10-3685 10-5828 10-8021 1 1 -0266 10 (a) 1-28008 1-34392 1-41060 1-48024 I-55297 1-62889 (b) 11-2034 11-4639 II-73I4 12-0061 12-2882 12-5779 15 (a) 1-44830 1-55797 1-67535 1-80094 I-93528 2-07893 (b) I7-93I9 18-5989 19-2957 20-0236 20-7841 21-5786 20 (a) 1-63862 1-80611 1-98979 2-19112 2-41171 2-65330 (b) 25-5447 26-8704 28-2797 29-7781 3I-37I4 33-o66o 25 (a) 1-85394 2-09378 2-36324 2-66584 3-00543 3-38635 M 34-1578 36-4593 38-9499 41-6459 44-5652 47-7271 30 (a) 2-09757 2-42726 2-80679 3-24340 3-74532 4-32I94 (*) 43-9027 47-5754 51-6227 56-0849 61-0071 66-4388 35 (a) 2-37321 2-81386 3-33359 3.94609 4-66735 5-51602 (*) 54-9282 60-462 1 66-6740 73-6522 81-4966 90-3203 40 ( fl ) 2-68506 3-26204 3-95926 4-80102 5-81636 7-03999 (*) 67-4026 75-40I3 84-5503 95-0255 107-030 120-800 45 (a) 3-03790 3-78160 4-70236 5-84118 7-24825 8-98501 (b) 81-5161 92-7I99 105-782 121-029 138-850 159-700 50 (a) 3-437 n 4-3839I 5-58493 7-10668 9-03264 11-46740 (*) 97-4843 112-797 130-998 152-667 178-503 209-348 0-0107238654 0-0128372247 0-01494703498 log of amount of ji at 2 '5 % at 3 % at 3-5 %. in I year 0-0170333393 0-0191162904 1-0211892991 at 4%- at 4-5 % at 5 %. 597 PRESENT VALUE TABLE (showing the present -value- of future payments of (a) i, (6) i yearly}. TIME RATES PER CENT. IN YEARS. 2-5 3 3-5 4 4-5 5 1 (a) 0-97561 0-97087 0-96618 0-96154 0-95694 0-95238 (b) 0-97561 0-97087 0-96618 0-96154 0-95694 0-95238 2 (a) 0-95181 0-94360 0-93351 0-92456 0-9I573 0-90703 (*) 1-92742 1-91348 1-89969 1-88609 1-87267 1-85941 3 (a) 0-92860 0-91514 0-90194 0-88900 0-87630 0-86384 (b) 2-85602 2-82862 2-80164 2-77509 2-74897 2-72325 4 (a) 0-90595 0-88849 0-87144 0-85480 0-83856 0-82270 (b) 3-76198 3-71711 3-67308 3-62990 3-S8753 3-54595 5 (a) 0-88385 0-86261 0-84197 0-82193 0-80245 0-78353 (6) 4-64583 4-57971 4-5I505 4-45182 4-38998 4-32948 6 (a) 0-86230 0-83748 0-81350 0-79031 0-76790 0-74622 (b) 5-50813 5-41720 5-32855 5-24214 5-I5788 5-07569 7 (a) 0-84127 0-81309 0-78599 0-75992 0-73483 0-71068 (b) 6-34939 6-23029 6-11454 6-00205 5-89271 5-78637 8 () 0-82075 0-78941 0-75941 0-73069 0-70319 0-67684 (*) 7-17014 7-01969 6-87396 6-73274 6-46321 9 (a) 0-80073 0-76642 0-73373 0-70259 0-67290 0-64461 (<5) 7-97086 7-78611 7-60769 7-43533 7-26879 7-10782 10 (a) 0-78120 0-74409 0-70892 0-67556 0-64393 0-61391 (*) 8-75206 8-53020 8-31661 8-11090 7-91273 7-72I73 15 (a) 0-69047 0-64186 0-59689 0-55526 0-51672 0-48102 (b) 12-3814 u-9379 11-5174 11-1184 10-7395 10-3797 20 (a) 0-61027 0-55368 0-50257 0-45639 0-41464 0-37689 (*) 15-5892 I4-8775 14-2124 J3-5903 13-0079 14-4622 25 (a) Q-53939 0-47761 0-42315 0-37512 0-33273 0-29530 (b) 18-4244 17-4131 16-4815 15-6221 14-8282 14-0939 30 (a) 0-47674 0-41199 0-35628 0-30832 0-26700 0-23138 (^) 20-9303 19-6004 18-3920 17-2920 16-2889 I5-3725 35 (a) 0-42137 0-35538 0-29998 0-25342 0-21425 0-18129 (b) 23-1452 21-4872 20-0007 18-6646 17-4610 16-3742 40 (a) 0-37243 0-30656 0-25257 0-20829 0-17193 0-14205 (b) 25-1028 23-1148 21-3551 19-7928 18-4016 17-1591 45 (a) 0-32917 0-26444 0-21266 0-17120 0-13796 o- 1 1 1 30 W 26-8330 24-5187 22-4955 20-7200 19-1563 I7-774I 50 (a) 0-29094 0-22811 0-17905 0-14071 0-11071 0-08720 (<*) 28-3623 25-7298 23-4556 21-4822 19-7620 18-2559 1-9892761346 log or present value at 2-t; y 1-9871627753 1-985096503 at 3%- at 3-5%. of I year 1-9829666607 7-98088370096 7-9788107009 at 4 %. j at 4-5 %. at 5 %. 598 ANSWERS PART I Exercises I. P. 4. 2. (a) 20, (b} 23, (c) 15, (d) 10, (g) 1 8, (/) 62. 3. (a) 86, (6) 100, (c) 101, (af) no, (,?) HI. 4. (a) 128, (3) 182, (c) 108, (/) 1 80. 5- (a) 303. (*) 330, (c) 333. 6. (a) 2006, (b) 6002, (<r) 6020, (rf) 6200. 7. (a) 8706, (3) 8076, (c) 876. 8. (a) three, (b} eight, (<:) eighteen, (d) eighty-one, (e) twenty-five, (/) fifty-two, () five hundred and twenty, (h) three hundred, (z) three thousand, (/) three thousand and five, (k) three thousand and fifty; (/) three thousand five hundred, (m) five thousand and three, (n) one hundred and eighty-six, (o) one hundred and sixty- eight, (/) eight hundred and sixty-one, (-7) eight hundred and sixteen, (r) seventy one, (s) seven hundred and one, (t) seven thousand and one. Exercises II. P. 6. I. 1914. 2. 3087. 3. 2748. 4. 2869. 5. 71. 6. 463- 7. 131. 8. 417. 9. 889. 10. 939- 11. 681. 12. 1453. 13. 1599. 14. 1911. 15. 517. 16. 1072. 17. 1950. 18. 2713- 19. 2002. 20. 1687. 21. 2814. Exercises III. P. 7. 1. Eighty. 2. One thousand. 3. Sixty. 4. Eight hundred. 5. Seven hundred 6. One. 7. One thousand. 8. Eight hundred. 9. Twenty. 10. Thirty. 11. Thirty. 12. Two. 13. Seven hundred 14. Three hundred. 15. Ten. 16. One. 17. Nought. 18. 8814. 19. 2312. 20. 2282. 21. 5016. 22. 765. 23. 508. 24. 1236. 25. 172. 26. in. 27. 712. 6oo A MODERN ARITHMETIC Exercises IV. P. 7. 1. 1311. 2. 1276. 3. 405. 4. 1719. 5. 660. 6. 1311. 7. 1172. 8. 1721. 9. 503. 10. 12377. 11. 2359. 12. 9972. 13. 3330. 14. 2121. 15. 7632. 16. 7529. 17. 2710. Exercises V. P. 9. 1. 450. 2. 411. 3. 149. 4. 417. 5. 334, Exercises VI. P. 11. 1. 1310. 2. 1655. 3. 1872. 4. 1562. 5. 2657. 6. 4950. 7. 13154- 8- 3172. 9. 5053. 10. 4684. Exercises VII. a. P. 14. 1. (a) 9, (J) 16, (f) 22, (d) 27. 2. () 9, (J) 8, (<:) IO, (d) II. 3. 7. 4. 9. 5. 5. 6. 13 inches. 7. 4 yards. 8. 12 Ibs. Exercises VII. b. P. 15. 1. (a) 917, () 956, (c) 149, (</) 1124, (e) 763, (/) 692, () 1147, (h) 8 5 08, (I) 2. 2. () 53, (*) 16, (c) 144, (<0 203, (*) 75, (/) 688, (g] 534, (h) 397, (0 1286, (/) 1693, (k) 1087, (/) 714, (m) 72, () 98, (o) 997- 3. 4892. 4. 4221. 5. 802. 6. 154. 7. 1978. 8. 929. 9. 1999. 10. 1468. 11. 3834. 12. 1926. Exercises VII. c. P. 16. 1. 24, 87, 4, 1 6, 4, i, i, 7, 13, 2, 280. 2. 20, 150, 13, i, 5, 7, 6. 3. 2, 24, 104, 40, 2, 172. 4. 44, 6, 10, 10, 52, 94. 5. 25, 140, 36, 13, 4, 2, 180. Exercises VII. d. P. 17. 1. 4- 2. 1 8. 3. 3. 4. 21. 5. 14. 6. 12. 7. 7- 8. 13. 9. 6. 10. 14. 11. 7. 12. 14. 13. 6. 14. 8. 15. 5. i6. 15. Exercises VIII. c. P. 21. 1. (a) 2 x 24, 4 x 12, 3 x 16, 6 x 8 ; (d) 2 x 14, 7 x 4 ; (^5x7; (d) g x 10, 3x30, 2x45, 6x15, 5x18; (e) 2x44, 4x22, 8xn ; (7)4x8, 2X 16. 2. (i) 14, (ii) 22, (iii) 99, (iv) 96, (v) 96, (vi) 91. ANSWERS 601 Exercises IX. a. P. 23. 1. () 12, (*) 9, (') 18, (d) 15, W 24. 2. () 96, (b} 72, (*) 48, (d) 60, M 108. 3. () 24, (b) 30, (c) 42, (d) 12, W 48. 4. (fl) 28, (b) 84, (*) 49, (d) 63. 5. 96. 6. 54- 7. 1 80. 8. 1 1 20. 9. (a) 50, (b) loo, (<r) 150, (d) 200. 10. () 75, (6} 125, (0 175- 11. () 5oo, (b} 1000, (f) 375, (<0 625, W 875- 12. () 432, (b} 576, (*) 1728, (d) 720, W 864. 13. (a) 472, (b) 1180, (*) 1652, (rf) 188* :, W 2124. 14. () I33i, (b) 363, W 726, (d) 605. 15. (a) in, (b} 222, (c) 333, (</) 148. 16. (a) 112, (b} 224, (f) 84. 17 . () 560, (b) 896, (c) 1008. 18. () 3x8, 4x6, 2 X 12 ; (b) 2X 16, 4x8; (0 3 x i5 ,5x9- Exercises IX. b. P. 24. l. 405. 2. 324. 3. 1380. 4. 1224. 5. 13440. 6. 3836. 7. 45760. 8. 2660. 9. 4183. 10. I3II. 11. 8091. 12. 6144. 13. 7992. 14. 23180. 15. I2OOI5. 16. 4838400. 17. 241902. 18. 491337. 19. 515080. 20. 330165. 21. 501160. 22. 383358. 23. 229914. 24. 242782. 25. 130400. 26. (a) 25740, (b) 42250, (c) 31916, (d) 560, () 29000. 27. (a) 388, () 227, (c) 214, (a?) 894, (<) 3510, (/) 3079, (g) 3561, (h) 176, (0 5173, (/) 3708, (k) 3048, (/) 2181, (m) 70, () 135, (0) 13, (p) 366, (?) 715. 28. 512616 Ibs. 29. 104390 miles. 30. n ozs. 31. 24 inches. Exercises IX. c. P. 26. 1. (a) 9360, (6) 936, (c) 468000, (d) 163800. 2. (a) 2679, (b) 267900, (r) 2679000. 3. (a) 4000, () 35000, (') 400000, (d) 6000, () 81000, (/) 2-800. Exercises X. P. 27. 1. (a) 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 ; (*). 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049. 2. 5, 25, 125, 625, 3125. 3. (a) 2 3 , 2 fi , 2 9 , 2 12 , 2' 5 ; (3) 2 4 , 2 8 , 2 12 , 2 2 20 . 4. i, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400. 602 A MODERN ARITHMETIC 5. (a) 3, (b) 4, (c) 5, W 6, (e) 7, (/) 8, (g) g, (h) 10, (t) n, (j 6. i, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728. 7. (a) 4, 9, 6, 5, o, i ; (3) 8, 7, 4, 5, 6, 3, 2, 9, o, i. 8. () Exercises XI. P. 28. 6. (a) 22$, 625, 1225, 2025, 3025; (b) 289, 484, 1089, 8281, 6724, (c) 10404, 41209, 93025. 7. (a) 9, (b) 16, 25, 36, 49, 64, 81, (c) 100. 8. 169. 9. (a) i 7 2-i6 2 ; (b) I5 2 -I4 2 5 (0 i6 2 -! 5 2 . 10. 5 rows 25 pennies. 11. 576. Exercises XII. a. P. 31. 1. 6, 10, 18, 24. 2. 7, 13, 32, 8. 3. 9, 18, 14, 32. 4. 15 boys, 4 apples. 5. 7, 10, 4, 40. Exercises XII. b. P. 31. 1. (a) Quotient 791, remainder 2; (b) quotient 475, remainder o (c} quotient 339, remainder 2. 2. 2001. 3. 149. 4. 250 feet. 2 inches over. 5. 178 times. 6. 546. 7. 7. Exercises XII. c. P. 32. 1. (a) 673, rem. 2; () 538, rem. 4; (<:) 449, rem. o; (d) 384, rem. 6; (') 336, 6;(/)299, 3; (g) 244, rem. 10 ; (h] 224, ,, 6. 2. (a) 2029, rem. I ; (*) 1893, rem. 12 ; (0 1775. r em. 7 ; (<0 1578, 3; () 1352, 15; (/) 1183, 15- 3. (a) II2I, 2; (*) 945, 29; (r) 864, ,, 29; (d) 8 4 0, 29; W 756, 29; (/) 687, 41. 4. (a) 25. , o; (^) 23, 21 ; w 22, 47; (d) 20, ,, 25; () 18, 45; (/) 17, 37. 5. (a) 38, 48; (^) 33, 12; w 29, w 26, , 545 (^) 25, 345 (/) 24, ," 96.' 6. (a) 24, 29; (b) 22, . 35; w 19, 77; (d) 18, 29; W 17. 5; 15, 119. Exercises XII. d. P. 33. 1. (a) u, rem. 5 ; (b) 4, rem. 14 ; (c) 6, rem. 4. 2. 6 groups, rem. 7. 3. (a) 5, rem. n ; (6) 3, rem. 14. 4. 5. ANSWERS 603 Exercises XII. e. P. 36. (a) (i) 2710 , rem. ii ; (ii) 2451 , rem. 2 ; (iii) 3835 , rem. II ; (iv) 3046 5 55 4; (v) 1502 , 16 (vi) 709, ,, 17; (vii) 3613 5 5, II. P) (i) 1113 , rem. 13 ; (ii) 1006 , rem. 39 (iii) 1575 , rem. 16 ; (iv) 1251 , 55 6; (v) 617 , 10 (vi) 291 , 28; (vii) 1484 , ,, 6. (c) (i) 506, rem. 103; (ii) 458, rem. 41 (iii) 717, rem. 25 ; (iv) 569, ,, 75; (v) 280, ,, 122 (vi) 132, 88; (vii) 675, 5, 85. (d) (i) 130, rem 201 ; (ii) 117, rem. 449 (iii) 184, rem. 264 ; (iv) 146, 274; (v) 72, 146 (vi) 345 72; (vii) 173, 55 416. w (i) "8, rem. 39M (ii) 107, rem. 200 (iii) 1 68, rem. 16 ; (ivj 133, 55 237; (v) 65, 5, 437 (vi) 31, 49; (vii) 158, 1 60. (/) (i) 149, rem. 208; (ii) 135, rem. 80 (iii) 211, rem. 229 ; (iv) 1 68, 5 , 6; (v) 82, ,, 368 (vi) 39, 61; (vii) 199, 5> 127. (<") (i) 85, rem 206 ; (ii) 77, rem. 88 ; (iii) 1 20, rem. 496 ; (iv) 95, 55 617; (v) 47, ,, 205 ; (vi) 22, 242 ; (vii) 113, 55 507. (*i (i) 67, rem. 567; (ii) 61, rem. 133 ; (iii) 95, rem. 626 ; (iv) 75, 5, 912; (v) 37, 448; (vi) 17, 650; (vii) 90, 55 130. Exercises XII. f. P. 36. 1. (a) 4, (b) 4, (c) 6. 2. (a] 5, (b) 4, (e) 4, (d) 2. 3. (a) 28, (b) quotient 25, rem. 7, (c) 866, (d) 1095, (e) 28, (/) 35. 4. 5 or o. 5. 25, 50, 75. 6. o, o, 25, 50, 75. 7. 5, 5, 12, 7, 8, 8, 3, i, 16. 9. 3150, 450, 3600, 1200, 8425. 11. 92, 15, 35, 77, 188. Exercises XIII. P. 38. 1. 2176, 51232, 6328; 1836, 3024. 2. (a) o, i, (3) 8, i, 21, (c) 7. 3. 59, and 182 Ibs. over. 4. 62nd. 5. <5 = 6, a=i. 6. 12852 and 17856. 7. Dividend = 83618, quotient = 352. 8. 28 and 62 inches over. 9. 29. 10. 8. 604 A MODERN ARITHMETIC Exercises XIV. P. 41. 1. 8. 2. 7 (to nearest unit). 3. 12. 4. 7811. 5. 55, 2 approx. 6. Between 63546 and 63545. 7. (a) 19980, (/$) 3092, (c) 8643. 8. 46. 9. Apr. 1 6, Sept. 20. 10. (a) 175706, (b) 591370, (<:) 1074. Exercises XV. P. 45. 3. 45. 4. (a) 36, (*) 28. 5. (a) 36, (3) 70, (c) 49, (<0 92, (*) 78, (/) 63. Exercises XVI. P. 46. 5. 9. 17. 18. 6338, 37i- 2. 85, and 27 over. 3. 79712. 4. 123. 28348. 6. 9359. 7. 3465. 8. 12288. 49. 10. 4616. 11. 4035230. 12. 19664. 56, 28 over. 14. 620. 15. 1512000. 16. 2019. One hundred thousand and two hundred millions, two hundred thousands one hundred. 7256. 19. 1 8. 20. 8620. 22. 120886698962. 23. (a) 69801000, (b) 64791000, (c) 75416000, (d) 72326000. Imports. Exports. Imports. Exports. 24. 1892 39326 37200 25. 1892 253076 243967 1893 38383 29685 1893 239698 192628 1894 41923 36933 1894 265134 265713 1895 44086 41247 1895 26I2I2 307787 1896 46444 34507 1896 275799 247410 1897 53820 36731 1897 276901 295728 1898 53521 44197 1898 302223 285769 1899 55778 49433 1899 319848 346415 1900 323818 242317 1901 336II8 266841 26. 27. (a) 224089, (b) (a) (1000) I6W 35308, (c) ?86. (IQO; 21197, (d) 18198, [) I4SQOQI. (1002^ () 15563, T6TTO7O. (/) 2I4IO. lf)\ VAO.IZA (c) 16701, (d) 98092, (e) 9788. 28. (a) 11225954, (b) 254334, (c) 1445210, (d) 398119, (e) 10489990, (/) 5767056, (g) 9185720. 29. (a) 19125355, (6) 18178835, (c) 15016677. 3. (a) 3741985 to nearest , (b) 481737 to nearest , (c) 4223721 (^) 1922352. ANSWERS 605 Exercises XVII. b. P. 55. 1. (a) yxl. t (b) 4 orf., (c) so/., (d) 6od., (e) 72*, (/) 8 4 </., (g) g6<t., (h) io&/., (*) I2o, (/) i32aT., () 1440?., (/) 156^., (w) i68dl, () l8o/., ((7) 192^., (/) 204</., (q) 2i6d., (r) 22&/. t (5) 240^. 2. (a) i. 55., (6) 1, (c) 2. is. Sd., (d) 2. ios., (e) 3, (/) 3- 105., () 3- 15^.. (*) 4, (0 4- 3J. 4^- Exercises XVII. c. P. 55. 1. (a) 895^., (6) iyo2d., (c) 11954^., (rf) I4i799^-, () 168965^., (/) 2321917^., (g) 1681348^., (h) 4085743^. 2. (a) 2920, (J) 575, (f) 8227, (rf) 282143, () 118521, (/) 77365, (g) 733883, (h) 412199. 3. (a) 1383, (6) 2345, (r) 23550, (<f) 386723, (e) 16001, (/) 77274, (g) 243607, (A) 47800. 4. (a) 20, (3) 30, (r) 45, (d) 91, (#) 147, (/) 141. 155., () ^774. 18^., (h) 281. i8s. 6d., (i) 1653. 4J. 6^. 5. (a) 15*. 9|flr., (*) ^3. sj. 9fflT., (r) ^13. 2f. O^M (rf) ^14. i&. orfl, (A) ^236. 65. orf. Exercises XVIII. a. P. 56. 1. (a) is. lid., (b) 3s., (e) 2s., (d) 45. 3*, () &., (/) 38^. 6t/. 2. (a) ifo., (b) aos. t (c} 45. 6d. t (d) ifo., () 205., (/) 75. Exercises XVIII. b. P. 57. 1. 415. 4?. i^. 2. ^"3073. I2J. lifl?. 3. 12030. 2s. i^ 4. ^78534. 165. 9^. 5. ^26861. ;j. &f. 6. ^18891. 75. orfl 7. ^23352. 3-y. 6|^. 8. 4081. 4*. ofrf. 9. ^37. uj. id. 10. ^573. 105. 3^. 11. ^68. 165. orf. 12. 106. 175. i^. 13. ^38. 85. u$d. 14. ^467. i2s. 6d., 18705. 15. ^119. 05. zd. 16. ^"4557. 145. od. Exercises XIX. a. P. 59. 1. (a) 3^., (<5) 9i</., ( f ) 15. 9jar., (rf) 35. 5^., () 65. 9jrf., (/) 105. 2. ( fl ) i5. iK, (*) if. 7i^, (f) 9*- iK (^) i4J. 3i<* () 165. (/) 205. \\d. 3. (a) 15. 3</., (*) 45. grfl, (0 75. 3^., (rf) 125. iiaT., (e) 17*. 3< 4. (a) 45. &/., (d) 65. &/., (f) 75. &/. 5. (a) 55. 4i<, (3) 85. a/., (^) 125. 4fflf., (</) 185. 606 A MODERN ARITHMETIC Exercises XIX. b. P. 59. 1. (a) S. is. iod., (b) 8. igs. 8f</., (c) i. 12*. 7d., (d) 19. is. 2. (a) 1907- I2s. yd, (6) 2427. i6s. 8d., (c) 346. igs. 2d., (d) 10528. is. gd. 3. (a) 12824. i&- o$d., (6) 9154- 7* iorf., (0 1519. 19* 2 I^ (0 63490. &. 9^- 4. (a) 2425. 185. 7f</., (b) 68547- J. 6*., (<) 22463. i6s. 4W- (*0 47i94i- 2s. 8d. 5. (a) 99864. 19* loiaT., () 310313. u. iQ\d., (c) 66928. u. ioj^., (d) 191125. I2J. IOif. 6- 59. its. lid. 7. 77. 8. 54. 15*. yd. 9. 4. 2s. od. gain. 10. 6517. 11. iSs. 2d. 12. 24. 13. 166. 4J. 9^. 14. The first is better by 139. 105-. ii</. 15. 45. 7</. in favour of payee Exercises XX. a. P. 62. 1. 3.5-. 9\d., 4*. 9%d., $s. S%d., Ss. 2d., 9s. Q\d. y 32 s. Sd., 36^. 9^. 47^. 3^-> 3- 25. SJrf. 2. 2s. 6d., 35-. 8^., 5^. iod., 6s. ^d., IDS. 4^., 19^. $d. t 31 j. ioa., 655. 76*. Sd. t logs. 3. 2j. 3^., 3^. 6a?., 4^. 9^., ss. 6d., 8j. 3^., 9^. 9</., 13^. 3^., 27 j. 6i/. 11. i8s. od. t 12. 4^. gd. 4. 2y. iifaf., 95. 2%d., los. od., us. 4^., 16*. 6%d., 22s. 2\d., 2gs. $d. 445-. SflT., 51^. 5^., 1145. lOflf., I2QJ. 8^. 5. 275. 6d., 3u. 9^., 51^. 7$d., s&r., 59^. 9^., 86^. yjdl, nu. 9^. 1235., 1455. 6d., 154^. 6^., 237.?. Exercises XX. b. P. 63. 1. 440. 195. 3^., 587. 19*. od., 734. iSs. gd., 881. iBs. 6d. 2. 2506. 61. 6|aT., 3222. 8^. si</., 3938. I05-. 31^., 1790. 4^. 8^ 3- 797- 7*. &*., 647. 175. sfrfl, 946. 17^. 10^., 697. I4J. 2irf. 4 - 14327- I7J. od-, 955 J - i8y. orf., 11939- 17-f- 6^., i353i- 17-^- 2c 5. 9281. u. ij^., 8437. 6s. stf., 12655. 19* 8i^f 16030. i&. 3^ 6. 146295. u. o^., 219442. lu. 6d., 292590. 2^. cw/., 365737. i2s. 6c 7 - i39Si. 5-r- &?., 15501. &. 4^., 16276. 95. 9aT., 24802. sj. 4r. 24220. 195. i\d. 8 - 52500. i2j. ck/., 65625. 151. o^., 70000. 16^. Oi/., 28711. y. 3f< ANSWERS 9. 181128. 13*. 3frf., 466869. us. 597186. uj. 9f^. 10. 585240. 195. 6d., 401244. 151. 2\d. 756675- 4*. <*/. 607 378995- 5*- 538687. 14-5-. 1. 3 6 - 19*- <** 4. 284. i6.r. nd. 7. 265. 3-r. 3d. 10. 473- 9-r- 3^- 13. 10570. 85. 9^. 16. 1141. 19*. od. 19. 306. 15^. n^, Exercises XX. c. P. 63. 2. 5. i2s. 6d. 3. 1148. 195-. lid. 5. 112. i3j. \\a '.. 6. 1147. 13^. i$d. 8. 174. 3-r. Oi/. 9. 75- 5J- 1\d. 11. 100. 12. 9036. iu. 3^- 14. i. IQS. sd. 15. 28427. ioj. Sd. 17. 17. '6s. Id. 18. 22. 125. Sd. 20. 37, 2s. 6d. Exercises XXI. a. P. 67. 1. is. 2d., &/., 7</., sJrf. ; ij. &/., is. 4d. t iod. 9 Sd., 2d. 2. icwT., 3^.; u. 9^., u. 3^., 5</. 3. 3j. id., 2s. 6d. t 2s. id., is. 6f^., iod. ; 4^. yd., 35. 8^., u. 8i/., lod. 4. pj. 8</., 6^. ojaf., 3^. o^. ; 135. 8Jrf., 9^. ij</., 65. laf. 5. 3. 6j. ioji/., 2. 13^-. 6d., 2. 4J. 7flf., i. i8j. 2^. and ^. ovtr, i. 135. 5^., i. gs. 8d. and i^. over, i. 2J. 3^. 6. 8, 10, 28, 38, 70. 7. 4 and 4^. over, 7 and 4^. over, 9, 12 and $d. over, 15 and 6df. over, 18. 8. 3. i8j. lid., 2. 16^. 4^. and i^d. over, 2. 3^. io^/. and id. over, i. 15^. iod. and 2^. over. 9- 36- 3J. 2^- and laT. over, 27. 2j. $</., 21. 13*. I id. and 2^/. over, 18. is. i\d. and i</. over. Exercises XXI. b. P. 67. 1. 43. 7-r. 2\d.^ 38. ios. iod., 33. oj. 8J^/. and i J</. over, 28. 185 2. 229. 155. 3i</. and 4^. over, 163. 7j. gd. and 1 1</. over, 141. ys. and 2^. over, 131. 5s. io%d. and Sd. over. 3- 363. 3J. lo%d. and u. 5frf. over, 306. Ss. io\d. and u. Sd. over, 280. 3-r. 6J^. and is. g\d. over, 262. 13*. 4^. and 4^. over. * 38. 7*. o^. and %d. over. 5. 83. 145. i\d. 6. 209. 155-. 8|^. 7. 437- r8j. 2^. 8. 790. or. gfrfl 9. 8063. 15^. oj^. 608 A MODERN ARITHMETIC 10. ,365. 13-r. 6d., undivided 5^. 6d. 11. ^259. 145-. g\d., remainder I s. $%d. 12. 69 times. 13. 8 ye 14. Man, ^3. is. 4^.; boy, i. los. Sd. 15. gs. *i\d. 16. 78. 17. 34 sums and i6s. Sd. remaining. 18. 108 children, 3s. gd. remaining 19. 235 coins. 20. 3054 rupees. 21. 58 of each. 22. (a) 4%d., $d. ; (b) Ss. 4d., is. ud., 3%d.', (c) i6s. Sd., 3 (d) 4- 3J- 4^. 19*. 2f<* 2J- 9^-; () 8. 6j. &*, ji. 18. &d ^s. $%d. ; (/) 41. 13s. 4d., g. 12s. 3$d., (g) ^83. 6s. Sd., ig. 4$. 7\d., 2. i4s. g^d. Exercises XXII. P. 1. 16. us. 6%d. 2. i. 145-. 6d. 3. 270. 4. ^"149. los. 4d. 5. ,29. Ss. 6d. 7. ^27. oj. gd. 8. 37. 7j. 3^. 6. 840 Ib. 9. 6. ss. od. 10. 109. 4s. od. 11. 9 + 48 or 57. 12. ^75. us. 3d. 13. 201. 4*. 6ar. 14. 1156 yd. 15. i. iSs. gd. 16. 15048 dollars. 17. 1344 Ib. 18. 2J. 6d. a yd. 19. 1st method by ^i. 13.5-. gd. 20. ?i. oj. lod. gain. 21. ^"3. 145-. gd. 22. Imports, ji i. I 3 j. ijif. ; exports, 6. 4*8 23. 10. i6j. 3d. 24. ^730. i6.r. od. 25. ^962. 26. ^68. 6s. Sd., g4- I 3 f. 7^-, ^87. 27. .242. 7^. lid. collected and ^257. 12s. i^. short. 28. ^i. 2s. 6d. 29. 3%d. 30. 2. 6s. lod. 31. ^3. 7J. uaT. 32. ^3. 3^.4^. 33. i. 2s. od. 3*. ^"65. 7s. 6d. or 523 half-crowns. 35. 10. 36. 66 payments, 62. 7.5-. 6d. left over. 37. ii payments, 43. 13^-. gd. left. 38. 3008 men. 39. 345, 7". iu. Zd. 40. 3. is. gd. 41. 6. 6s. 6d. each, i. los. od. left. 42. jio8. 45. 2%d. 43. ^"2. 7*. 3^. 44. 516. Exercises XXIII. a. P. 76. 1. (a) l ft. 4 in., (b) 2 ft. 2 in., (c) 3 ft., (aT) 4 ft. 2 in., (*) 8 ft. 4 i (/) 1 6 ft. 8 in. 2. (a) 29, (3) 63, (0 360, (d) 312, (*) 3600, (/) 364. 3. (a) 5280, (b) 660, (*) 1320, (rf) 7920, (e) 660. ANSWERS 609 Exercises XXIII. c. P. 77. 1. 63360. 4. 2 10 yd. 2 ft. 7. 96 yd. 2 ft. 4 in. 10. 5 m. 867 yd. I ft. 13. 22 ft. 6 in. 2. 5280. 3. 30. 5. 7m. 271 yd. I ft. 8 in. 6. 1500. 8. 8 m. 1593 yd. I ft. 9. 266 yd. 2 ft. 11. 261 ft. 12. 25416. 14. 41 yd. i ft. o in. 15. 11 m. 440 yd Exercises XXIV. P. 81. 4. 7. 10. 12. aw jrus. i IL. 11 295- . 9 yd. o ft. 7 in. 2 m. 1361 yd. i (a) 366 ft. o in. (e) 6 ft. o in., 17 ft. 6 in., 73ft. oin., (/) 35ft. oin., i ft. 3 in., 17 ft. o in., II ft. 6 in., 111. m, 4 in. 11*3 yu. 5. 105 yd. I ft. 8 in 8. 252 yd. 2 ft. 9 in ft. 11. 3 m. 1608 yd. ; (b) 304 ft. 10 in. ; (c) 198 3 ft. o in., i ft. o in., 6 ft. oin., i ft. c in., 49 ft. 9 in., o ft. 10 in., 218 ft. 6 in., 10 ft. 6 in., 3 ft. o in., 10 ft. ii in., 71 ft. 2 in., 12 ft. 6 in., i ft. o in., 37 ft. oin., o. lyu. 6. 678 yd. 8 9. 107 yd. i ft. ; (d) 80 ft. i in. ; 2 ft. o in., o ft. 23 ft. i in., 19 ft. 7 ft. o in., 40 ft. 40 ft. oin., 2 ft. 7 ft. 7 in., i ft. 49 ft. ii in., 8 ft. 63 ft. 4 in. in. ft. 6 in., 9 i n -> O in.; o in., 6 in., o in., Exercises XXVI. b. P. 86. (A) 1. 60 mm. 2. 600 mm. 3. 6000 mm. 4. 6060 mm. 5. 6600 mm. 6. 6660 mm. 7. 7300. 8. 7003. 9. 303. 10. 52. 11. 18000. 12. 1010. 13. 1120. 14. 6020. 15. 6002. 16. 4200. 17. 4320. 18. 6006. 19. 7080. 20. 8020. 21. 8032. 22. 9091. 23. 2029. 24. 238. 25. 16803. 26 - 5302. 27. 9027. (B) 1. 7 cm. 6 mm. 2. 7 dm. 6 cm. 3. 7 dm. 6 mm. 4. 8 m. 2 dm. 6 cm. 5. 9 dm. 8 cm. 6. 5 m. 3 dm. 9 mm. 7. 23 m. 4 dm. 8. 9 m. i dm. 2 mm. 9. 9 dm. i cm. 2 mm. 10. 9 m. i cm. 2 mm. 11. 8 m. 7 dm. i cm. 6 mm. 12. 8 m. 7 dm. 6 cm. 13. 8 m. 7 dm. 14. 9 dm. 6 cm. 2 mm. 15. 83 m. 4 dm. 5 cm. J.M.A. 2Q 6io A MODERN ARITHMETIC 16. 8 m. 3 dm. 4 cm. 5 mm. 17. 8 m. 4 cm. 6 mm. 18. 7 m. 6 dm. 2 cm. 3 mm. 19. 9 m. 2 dm. 6 cm. 20. i m. 7 dm. 6 cm. 3 mm. (c) 1. 4 dm. 2. 8 mm. 3. 3m. 4. i decametre. 5. 2 dm. 6. 3 cm. 7. 2 dm. 8. 8 dm. 9. 7 cm. 10. 30 m. 11. 8 m. 12. 3 dm. 13. 70 m. 14. 6 dm. 15. 70 m. 16. 5 cm. 17. i m. 18. 10 m. 19. I dm. 20. 70 m. 21. 7 m. 22. 2 cm. Exercises XXVII. P. 87. m. dm. cm. mm. m. dm . cm. mm. m. dm. cm. mm. 1. . o . 2 . 2. . 2 . O . O 3. 2 . . o . o 4. i . 2 . O . O 5. 2 . 3 . O . 6. 2O . 5 o . 7. 44 O . O . o 8. 44 . . O . O 9. 4 . 4 o 10. 23 2 . . o 11. I . 7 . o . o 12. 34 5 o . 13. 131 . 3 O . o 14. O . i . . o 15. o . . i . 16. o . O . O . I 17. . 3 . 2 18. o . 2 . 3 3 19. . 2 . 3 o 20. o . 2 . . 3 21. o . 2 . 3 22. 10 . 6 . i . 2 23. I . . 6 . 7 24. 2 . 4 i . 3 25. 5 7 - 4 26. 8 . 8 . 8 . 4 27. O . 7 . 6 Exercises XXVIII. P. 89. 1. 3. 5. 3-28 metres. (a) 70-25 dm., 0-296 m. 2. (a) 4-16 dm., (b) 702-5 cm., (c) 7025 6. 13-236 m. (b) 41-6 cm., (c) 416 mm. 4. 14-069 m. mm. 7. (a) 847-5 dm., (b} 8475 cm., (c) 84750 mm. 8. 71-8. 9. 7-1- 10. 0-7. 11. 31-7- 12. 3-02. 13. 6-2. 14. 3-202. 15. 4-003. 16. 37-03. 17. 20-02. 18. 300-4. 19. 230-2. 20. 3-6. 21. 412. 22. 1-8. 23. 4160. 24. 6020. 25. 60-2. 26. 61-24. 27. 1-2. 28. 412. 29. 3-024. 30. 1260. 31. 4 cm. 32. 2 mm. 33. 300. 34. 3000. 35. 4- 36. 50. 37. 300. 38. O-I. 39. o-obi. 40. 2. 41. IOO. 42. IOO. 43. 500. 44. IOOO. 45. IOOO. 46. 20. 47. 200. 48. 250. 49. 4000. 50. 2OO. ANSWERS 611 Exercises XXIX. P. 92. m. dm. cm. mm. m. dm. cm. mm. m* dm. cm. mnii 1. 6 . 6 . 6 . 5 2. 15 7 7 9 3. 29 . 4 . 4 . 6 4. 23 2 . 6 . 5 5. 38 2 . 5 6 6. 26 . i . 4 . 8 7. 137 . 2 . 6 8. 51 . 8 . 2 . i 9. 35 I . 8 . 6 10. 40 . I . 4 7 11. 81 . . 8 . 6 12. 84. 3 . . 7 13. 89- 2 . 7 i 14. 20 . I . 5 5 15. 7 9 . o . 8 16. . 7 . 2 . o 17. i . 2 . 3 7 18. 4 3 7 o 19. 76 . 6 . 2 . 9 20. 96 . 3 i . 3 21. 2 . 8 . o . 22. 5 7 6 . 5 23. 20 . o . 6 . 8 24. 3 7 7 9 25. i . 5 - 3 7 26. 2 . 4 . i 27. . 2 7 o 28. 4 5 8 . o 29. IO . 5 6 . 8 30. 3 7 7 2 31. o . 9 4 4 32. . 9 . 9 33. 5 2 . i . I 34. 6 . 5 8 . 9 35. 41-161. 36. 38-55 tons. 37. 73-62 ft. 38. 9-11 ft. 39. 28- 1 7 gallons. 40. (i) 22-424, (ii) 506-309, (iii) 430-08, (iv) 191-851, (v) 367-537, (vi) I9I749, (vii) 5 22> 879 (viii) S' 2 ** ( ix ) 70-182, (x) 47-261, (xi) 185-78, (xii) 99-91, (xiii) 15-28, (xiv) 100-84, (xv) 22-83, (xvi) 151-717, (xvii) 94-075, (xviii) 189-27, (xix) 108-723. Exercises XXX. P. 96. 1. 65 m. 8 dm. 7 cm. 2 mm. m. dm. cm. mm. m. dm. cm. mm. m. dm. cm. mm. 2. (i) 52 . o . 2 . 4 (ii) 1 10 . 9 . 2 . 5 (iii) 90 . 5 . 9 . 6 (iv) 86 . 3 . 8 . 2 (v) 1.5.0.4 (vi) 26 . o . 4 . 3 (vii) 96 . 2 . 6 . 4 (viii) 69 . o . o . o (ix) 73 . o . o . o (x) 47 . i . o . o (xi) 33 . 5 . o . o (xii) 121 . 5 . 3 . 6 3. 6 m. 2 dm. o cm. 6 mm. m. dm. cm. mm. m. dm. cm. mm. m. dm. cm. mm. 4. (i) 0.5.2.7 (ii) I.5.9-5 ("i) 0.3.4.9 (iv) 0.8.9.6 (v) 0.8.2.0 (vi) 2.3.7.7 (vii) 0.9.9.3 (viii) 1.1.8.9 (ix) 0.3.2.8 (x) 4.0.5.7 (xi) 0.7.5.0 (xii) 7.0.7.8 (xiii) 4.1.4.1 (xiv) 0.4.1.5 (xv) 0.7,4.8 6l2 A MODERN ARITHMETIC 5. i -35 ft, 1 6-40 ft. 6. (i) (a) 26-883, (6) 44-805, (r) 62727, (d) 80-649 ; (ii) (a) 0-125, (b) 0-250, (r) 0-375, (d) 0-5 ; (iii) (a) 98-112, (b) 126-144, (<r) 112-128; (iv) (a) 12-648, (3) 9-486, ( f ) 22-134, (</) 28-458. 7. (i) (a) 3-8, (3) 1-9, (c) 0-95; (ii) (a) 0-0207, (6) 0-0069; (iii) (a) 4-543, (6) 1-947; (iv) (a) 124-875, (&) 208-125, (*) 69-375; (v) () 0-5, () 0-25, (r) 0-125; (vi) (a) 0-75, (d) 0-375; (vii) (a) 102-816, (J) 137-088, (c) 34-272; (viii) (a) 0-0008, (b) 0-0006, (f) 0-0004, (d) 0-0003, (^) 0-0002 ; (ix) (#) 0-20594, () 0-10297; (x) (a) 18-0012, () 9-0006, (f) 4-5003. (xi) (a) 1-44, () 0-6, (c) 0-9, (</) 0-8. 8. (a) 0-7878, () 3-054 lb., (c) 28-4691 ft. 9. 1776. 10. 9-98 lb. Exercises XXXI. a. P. 99. 1. (a) 4000, 40, 400; (b) 9000, 9, 0-9, 150, 18; (c) 100, 50, 5, 2, 3000, 300 ; (d) 40, 4, 4000, 56, 720 ; (*) 0-5, 5-0, 15-0, 0-0005 ; (/) 3-72, 0-0372, 0-372 ; (^) 0-48, 1-44, 14-4, 0-144. 2. 1-284111. 3. 5-824 m. 4. 151-36 m., 227-04 m. Exercises XXXI. b. P. 101. 1. 694-375 lb. 2. 10 lb. 3. 64-1875 lb. 4. 482-5 lb. 5. 355-25. 6. 242-52 ton. 7. 3223-68 m. 8. 693 francs. 9. 174-8565. 10. 21-33859 lb. 11. 1 1 3-03859 grains. 12. (a) 88-375 francs, (b) 338-855 francs. 13. 40-586 lb. 14. 0-4192 c. ft. 15. o-ii6:6c. ft. 16. 6-72 c. ft. 17. () i6ft. 4-85 in., (b) 9 ft. 10-8974 in., (c) 10 ft. 5-984 in., (d) 9 ft. 10-18874 in., (e) 118 ft. 1-43811 in., (/) 2 ft. 6-834584 in., (g) 3 ft. 0-149534 in., (A) 3 ft. 4-011731 in., (i) 7 ft. 97006 in., (/) 10-657459 in., () i ft. 11-169245 in., (/) 3-984244 in. Exercises XXXI. c. P. 102. 1. () S539-38, (*) 553-938. 2. (a) 6-788396, () 67883-96, (r) 67-88396, (d) 2518-8522, (e) 251-88522, (/) 283147-57, (g) 2831-4757. ANSWERS 613 3. (a) 8-19434, (b) 81-9434, (c) 819-434, (d) 55-165, (e) 551-65. 4. (a) 37807-5, (b) 378-075, (<) 3-78075, (d) I397-3652, (<) I39-73652. 5. (a) 41499, () 414-99, (<) 4I49-9, (d) 78300, (*) 7830, (/) 0-783. 6. (a) 8720, (b) 872, (<) 0-872, (</) 117720, (e) 1177-2, (/) 11772. 7. (a) 19109-275, () 1910-9275, (c) 191-09275, (af) 1-9109275, (<?) 1618-668, (/) 16-18668. 8. (a) 5824-7424, (6) 592-704, (0 0-614125, (d) 5-359375- 9. (a) 254-7216, (b) 6-0025, (c) 0-0027648. 10. (a) 104-04, (b) 1092-727, (<r) 10100-25, (d) 250100-01. 11. (a) 803174-44, (b) 65-69202, (<) 893-62. 12. (a) 999998-56, (3) 9999-9375. (') 6399'99- Exercises XXXI. d. P. 103. 1. 1-8. 2. 1-6. 3. 500. 4. 0-00006. 5. 4-9. 6. 0-064. 7. 0-000004. 8. 640. 9. i. Exercises XXXI. e. P. 104. (b) (i) 0-25, (ii) 0-40, (iii) 0-51, (iv) 0-76, (v) 0-91. (c) (i) 0-39, (ii) 0-71, (iii) 0-36, (iv) 0-59, (v) 0-83. (a) 0-8, (b) 2-43, (c) 0-13, (d) 0-27, (e) 1-26. Exercises XXXII. a. P. 109. 1. (a) 10, (b) 0-8, (0 5, (a 7 ) 0-2. 2. (a) 3-2, () 0-32, (c) 20, (a?) 200. 3. (a) 3, (a) 40, ( f ) 0-4, (a 7 ) 80, (*) 0-08, (/) 0-6. 4. () 40, (b) 4, (f) 0-4, (d) 400, (*) 2000, (/) 0-3. 5. (a) 7-5, (d) 0-75, (c) 0-4, (aT) 6, (e) 60, (/) 50, (^) 0-05. 6. (a) 6, () 0-6, (c) 0-8, (a?) 4, (e) 400, (/) 20, (^) 2, (A) 0-2. 7. (a) 16, (*) 10, (f) 8, (of) 0-6, (*) 6, (/) 0-3. 8. (a) 3, (3) o-i, (c) 8, (a?) 10, (e) 9, (/) 30, (^) 0-2, (h) 2. 9. (a) 1-6, (b) o-i, (r) .0-08, (d) 0-8, () 0-02, (/) 2, (^) 0-4. 10. (a) 0-8, (b) 12, (r) 60, (d) 3, () 0-2, (/) 0-02, fc) 16, (A) 160. 11. (a) 20, (d) 200, (f) 16. 12. (a) 6-25, (b) 9, (r) 20. 13. (a) 8, (<5) 4, (c) 16, (a?) 32. 14. (a) 0-5, (J) 15, (^) 25. 6, 4 A MODERN ARITHMETIC Exercises XXXII. b. P. HO. 1. 7-76. 2. 26-923 c. in. 3. 1-0936 and 0-9 1 44. 4. 163-48. 5. 43 and 1-506 pints. 6. 9 rods and 5-67 in. over. 7. 2 m. and 5-26 in. over. 8. 2-34 approx. 9. 13. 10. 8. 11. (a) 47, (t>) 47, (c) 0-0047, (d) 0-00047, () 47- 12. (a) 9-3, () 9-3, (r) 0-093, (d) 0-000093, () 93ooo. 13. (a) 840, () 0-84, (e) 840, (</) 0-000084, () 0-84. 14. (a) 623, (0) 0-000623, (c) 0-00623, (aT) 623, (e) 0-0623. 15. (a) 735-6, (b) 7356, (') 7356oooo, (rf) 0-000007356, (e) 0-00007356. 16. (a) 95. (*) '095> (0 0-095, (aO 9-5. (*) 0-00095. 17. (a) 795-o6, (*) 795-6, (0 0-000079506, (d) 0-79506. 18. (a) 0-040075, (6) 0-040075, (c) 0-40075, (d) 4007-5, (e) 400-75. 19. (a) 80-963, (3) 0-080963, (r) 8-0963, (d) 8-0963, (*) 80963. 20. (a) 4-8596, (*) 0-048596, (c) 4-8596, (rf) 0-048596, () 48596. 21. (a) 0-056, (3) 5600, ( f ) 0-056, (aO 560. 22. (a) 708-5, () 0-07085, (t) 0-007085, (</) 7085000. 23. (a) 753-8, (3) 0-07538, (0 75-38, (d) 7538, (*) 75380. 24. (a) 3854-06, () 0-385406, (c) 38-5406, (</) 0-0385406, (*) 3854-06. 25. (a) 0-0973, (*) 973, (f) 973, (^) 97'3 () 973- Exercises XXXIII. P. 114. 1. 0-28775. 2. 0-479. 3. 0-951. 4. 0-533. 5. 0-412875. 6. 0-8715. 7. 0-40075. 8. 0-847. 9. 26 fur. o ch. 80 Iks. 10. 2 fur. 6 ch. 27-2 Iks. 11. 8 fur. 4 ch. 96 Iks. 12. o fur. 7 ch. 12 Iks. 13. 17 fur. 3 ch. 4 Iks. 14. 34 fur. 8 ch. 88 Iks. 15. 57 fur. o ch. o Iks. 16. 3 fur. o ch. o Iks. 17. o fur. o ch. 16 Iks. 18. 54 fur. 4 ch. 80 Iks. 19. 5 days 3 hrs. 20. 12. 21. 58-7 mis. per hr. 22. 160-38 yds. 23. 0-8 miles. 24. (i) 5 ft. 4-4 in., (ii) 5 ft. 0-8 in., (iii) 5 ft. 4 in., (iv) 5 ft. o in. (v) 5 ft. 3-2 in., (vi) 4 ft. 9-2 in., (vii) 5 ft. 1-2 in. 25. (i) 671112000, (ii) 8949600, (iii) 66960, (iv) 1040-4, (v) Sol. 26. 8 min. 20 sec. 27. 738 decametres. ANSWERS 615 Exercises XXXIV. P. 116. 1. 4sq. in. 2. 8 sq. cm. 3. 6 sq. in. 4. 1804 sq . mm. 5. 12 sq . cm . 6. 15 sq. cm. 7. 31 sq. cm. 8. 1 20 sq. cm. 9. 2772 sq. cm. 10. a x b. Exercises XXXV. a. P. 120. 1. II sq. yd. 4 sq. ft. 80 sq. in. 2. 27 sq. yd. 3. 31-4914 sq. cm. 4. 71-2 sq. in. 5. 65 sq. yd. 3 sq. ft. 6. 39 sq. yd. 7 sq. ft. 136 sq. in. 7. 2 sq. ft. 104 sq. in. 8. 300 sq. ft. 9. 359 sq. ft. 48 sq. in. 10. loo sq. ft. 11. 1-0124 acres nearly. Exercises XXXV. b. P. 121. (i) 2, (ii) 12, (iii) i, (iv) 25, (v) 36, (vi) 3, (vii) 50, (viii) 24, (ix) 40, (x) 10-5. 1. 350. 2. 700. 3. 280. 4. 4900. 5. 560. 6. 350. 7. 350. 8. 280. 9. 420. 10. 1400. 11. 8400. 12. 210. Exercises XXXV. c. P. 122. 1. ist case, 78 ft. ; 2nd case, 90 ft., 30 sq. ft. wasted. 2. 128 ft. 3. 30 flagstones. 4. 195 sq. ft. 5. 400 slates. 6. 936 slates. 7. 720. 8. 48 sq. ft. 9. 572 sq. ft. 11. 8 sq. in. 12. 744 sq. ft., ,18. I2s. od. 13. 540 sq. ft., ^"13. los. od. 14. 5 sq. ft. 15. 6 sq. ft., 4*. 6d. 16. 12 sq. in. 17. 440 sq. ft. 18. 220 ft. 19. i. 195-. od. Exercises XXXV. d. P. 125. 1. I sq. ft. 18 sq. in. 2. 88 sq. in. 3. 52 sq. ft. 4. ^"42. I2s. 6d. 5. 1 60 yd. 6. (a) 1 550 sq. in., (b) 10-76, (c) 1-196 sq. yd. 7. (i) 20 m. i dm. 2 cm., (ii) 4048-144 sq. m. 8- 37293 sq. ft. 9. 976. 10. Carpet, 239-25 sq. ft.; margin, 83-75 sq. ft. 11. 5-25 m. 12. 3150 cm. 13. 1 68 sq. ft., 108 sq. in. 14. 27 sq. ft. 72 sq. in, 15. (a) 414 ft., (b) 342ft., (c) 304 ft. 16. 10 ft. 616 A MODERN ARITHMETIC Exercises XXXVI. P. 128. 1. I55xio 2 sq. m. 2. 620 ac. 3. 1385 sq. m. 4. 227 sq. m. 5. 156 sq. m. 6. 490 sq. m. 7. 275 sq. m. 8. 165 sq. m. Exercises XXXVII. a. P. 131. 1. (a) 23 fr. 20 c., () 193 ft- 6o c -> M 8 3 & 7o c., (rf) 9 fr. 36 c., () 1945 fr. 30 c., (/) 876 fr. 20 c., () 897 fr. 60 c., (k) I fr. 76-23 c. 2. (a) 6 lires 30 c., () 7 lires 83 c., (<r) 9 dol. 30 c., (</) 3 dol. 75-6 c., (e) 8 dol. 21 c., (/) 356 drachmas 80 leptas, (g) 932 M. 13 pf., (h) 13 M. 2 pf. 3. (a) (i) I dol. 50 c., (ii) 2 dol. 60 c., (iii) 103 dol. ; (b) (i) 150 fr., (ii) 120 fr., (iii) 100 fr. ; (c) II dr. 20 leptas ; (d) (i) 1824 lires, (ii) 320 lires ; (e) (i) 515 M., (ii) 101 M. 50 pf. Exercises XXXVII. b. P. 131. 1. 76 c. 2. 4 fr. 70 c. 3. 131 fr. 10 c. 4. 89 lire 90 centesimos. 5. 307 dol. 20 c. 6. 2353 M. 86 pf. 7. 75 M. 76 pf. (approx). 8. 1200 dol. 150. 9. 48384 fr. o c. 10. 14 dol. 18 c. 11. 326 fr. 12. 101412 lires 48 c. 13. 3301 dol. 65 c. 14. 7 fr. 60 c. 15. i kr. 5 M. 90 pf. 16. 76 kr. 6 M. 71-6 pf. 17. 6 drachmas 80 leptas. Exercises XXXVIII. a. P. 133. 1. 24 cub. in. 2. (a) 12, (b) 6, (c) 40, (d) 432, (e) 144. 3. 48, 24, 12, 80, 864, 288. 4. 84, 42, 280, 2924, 1008. 5. (a) 192 c. in., (b) 72 c. in., (c) 576 c. in., (d) 360 c. ft., (e) 375 c. in., (/) 6300 c.c., (g) xxyxz c.c., (h} ax&xc c.c. 6. 1728 c. in., 27 c. ft. 7. (a) 1000 c.c., (6) loooooo c.c. Exercises XXXVIII. b. P. 134. 1. 121*5 c. in., 5760. 2 - 2835 c. ft. 3. 17600 c.c. 4. 2 c. ft. 1086 c. in. 5. i ft. 4 in. 6. 0-025 cm. 7. 207-125 c. in. 8. 14 ft. 9. 13 ft. 8 in.; volume, 1476 c. in. 10. 75. 11. Content, 1800 c. cm.; substance, 1080 c. cm. 12. 3-36 sq. yd. 13. 57200. 14. 180000 c. ft., 11250000 Ib. 15. 0-002 ft. 16. 70. 17. 45360 tons. 18. i c. ft. 1512 c. in. ANSWERS 617 Exercises XXXIX. b. P. 140. 1. 2079 bottles. 2. 7 gal. 3 qt. i pt. 3. i c. ft. 5 c. in. 4- 3'33 ft- 5. 5-12 in. 6. 72600 gal. 7. 59-319 c. in. 8. 5-05568 c. in. 9. 7 pt., 2-16 c. in. over. 10. 26fg4f3i3m. 11. i pt. 14 f 4 f 3 38-4 m. 12. 28 f 5 3 f3 45 rn. 13. A, 0-421 litre ; B, 0-239 litre ; C, 0-356 litre. 14. 0-864 in. Exercises XL. b. P. 145. 1. 67 ton 6 cwt. 2 qr. 2. 7 ton 12 cwt. 3 qr. 3. 3 qr. gib. looz. 4. (a) 482 lb., (b) 1988 lb., (c) 2173 lb., (d) 31433 lb., (e) 479 lb., (/) 1777944 !b., () 43601 lb. ; sum, 829 ton 10 cwt. o qr. 20 lb. 5. (a) 84 lb. I oz., (b) 63 cwt, (c) 8 ton 10 cwt. 3 lb., (d) 3 qr. 15 lb. 9 oz., (e) 27 cwt. 2 qr. 20 lb. i oz. 12 dr., (/) 86 ton 13 cwt. 6. i ton. 7. 105007 c. ft. 8. 99750 c. ft. 9. 102144 c- ft- 10 - 382 cwt. 2 qr. 11. 787-39 ft. 12. 57 ton 8 cwt. 3 qr. II lb. 14 oz. 13. 0-42 cm. 14. 26 weights; 1-92 lb. over. 15. 40 litres ; 0-18 gal. left. 16. 4204 ft. 17. (a) 6720 kilo., (b) 19 cwt. 2 qr. 16 lb., (c} 245 kilo., (d) 3 cwt. I qr. 17 lb. 1 1 -2 oz., (e) 3 lb. 6-208 oz. 18. i ton 17 cwt. 3qr- 1 8 lb. 14 oz. ton. cwt. qr. lb. oz. ton. cwt. qr. lb. oz. 19. (a) 8 . 14 . o 23 4 w 13 . 18 . 2 . 26 . o (c) 27 17 . i . 24 . o (d} 48 . 15 . 2 . 7.0 (e) 87 . 2 . o . 8 . 8 (f) 365 . 16 . 2 . 24 . 8 (g) 566 3 . I . 27 4 20. I ton 1 1 cwt. o qr. 3 lb. 8 oz. 21. 5 lb. i oz. ton. cwt. qr. lb. oz. dr. 22. (a) 88 . 5 . I . 22 . 4 . 8 (*) 64 - 3 3 23 13 . 7 + 3 (0 20 . 3 . 2 . 3 7 . 14 + 6 (d) 7 18 . 2 . 21 . 8 . 2+14 (e) 3 8 . 2 . 6 . 13 . 8+II2 (/) i . 18 . . 19 . 3 . 15 + 306 lb. oz. dwt. gr- lb. oz. dwt. gr. lb. oz. dwt. gr. 23. (a) 3 9 II . II (*) 5 . 3 . 16 . i (c) 34 . 7 . 12 , , 12 (d) 10 . o . 6 . 6 w 17 . 9 . 5 . 15 (/) 13 . o . 15 , , IO 24. (a) 14-08 drs., (b} 15 oz. 0-64 dr., (c) 16 lb. I oz. 1472 dr., (d) 49 lb. O oz. 12-544 dr. 6:8 A MODERN ARITHMETIC Exercises XLI. P. 147. 1. 3 min. 56sec., 23hr. 55min. 4Osec. 2. 365 days 5 hr. 48 min. 47 sec. 3. 0-2545, 0-2564, 0-2459, 0-2438. 4. o hr. 45 min. appro*. 5. 07784 days. 6. 0-089 ^ a 7 s - ? 2 9 da Y s I2 hr - 44 min. 9. (a) i, () 1-102, (r) 1-23, (rf) 1-38, (e] 1-62, (/) 2-06, (g) 3-36. Exercises XLII. P. 149. 1. 122-836123 and 77-163877. 2. 9-98. 3. ^o. 17^. 6d. 4. 2050. 5. 8 gal. 3 qt. i pt. 6. 5-75. 7. 10 Ib. i oz. 8. 2220. 9. 59-8. 10. 32. 11. 6d. 12. 53. iy- od- 13. 19-8219661 336 and 19-822. 14. 23780000. 15. 10. 16. 3-8 Ib. 17. 2061, I qr. left. 18. (a) 2. us. Sd., (b) 0-3614. 19. 45-168. 20. (a) 7224. us. od.; 40-75. 21. 40 sq. yd. 4 sq. ft. 24 sq. in. 22. 10065, '375- 23 - 2 7- 24. (a) 0-67, (b} 18-70, (c) 5-78, (d) 19-2, (e) 67-225, (/) 0-31, (g) 14-085, (h) 4-115, (i) 16-2, (/) 67-62. 25. 6-25. 26. 126-72. 27. ji2i. oj. 4^. 28. 5-4198. 29. (a) 59-04, 64-8, 40-3, (b} 637, 20-9, 22-93, (') 579* 2o8 5> I2 95> (^) 300, 9-84, 6-71, (e) 30 min., (/) 1-41. 30. (a) 202, (6) 30, 0-3. 31. 30-15 inches of mercury. 32. 115 sq. ft. 72 sq. in. 33. 1134. 34. 846,0-0046875. 35. 2340. I2s. ^\d. 36. 54-3. 37. (a) 3-58 in., (b) 3-17 in. 38. Hull 59-9, Oxford 6i-3. 39. (a) 1-75, (*) 3-68, (c) 3-94, (d) 2-94, () 2-84, (/) 1-68. 40. 22 ft. per sec. 41. 48-6261 sq. in., 313-7055 sq. cm., 6-45. 42. 7 mm. 43. 985-6. 44. 13 ft. 45. i. i-js. $d. 46. 1 6 hrs. 40 min. 47. 0-279464. 48. 275 sq. ft. 72 sq. in.; 6. i"js. yd. 49. 9 yds. i ft. 8-1672 in. 50. 50 ton 17 cwt. 3 qr. 12 Ib. 51. (a) 290, (b} 0-25196, (c} o. i;j. 3-35^. 52. ^o. 45-. &/. 53. 12 hrs. 5 min., II hrs. 56 min. Exercises XLIII. a. P. 158. 1. 4d. 2. Sd. 3. gd. 4. 12*. 6 5. is. gd. B. i. i2s. od. 7. 8 in. 8. i ft. 3 in. 9. zs. i\d. 10. 6 oz. 11. 104 Ib. 12. i cwt. 68 Ib. 13. 2. 4*. orf. 14. i stone. 15. 78 Ib. 16. 90 in. 17. iSs. 18. 100 yds. ANSWERS 619 Exercises XLIII. b. P. 158. 1. 2 cwt. 16 Ib. 2. 6. i6s. 6d. 3. 14. 7.5-. id. 4. 2 ton 10 cwt. 3 qr. 21 Ib. 5. 2 Ib. 10 oz. 10 dr. 6. 24 mis. i ch. 7. 63 yd. 2 ft. 8. 6 m. 5 dm. 3 cm. 2 mm, 9. 2144 gms. 10. 21 ac. 2 ro. 16 p. 11. 121. 12. us. $d. 13. 48 yd. 2 in. 14. 100 ton 13 cwt. 87 Ib. Exercises XLIII. c. P. 159. 2. (a) 14; (f 3. ILT. II 6. 3 9 yd. Exercises XLIV. P. 161. f^\ 726 W TV', \ .3.4L.-L 3 - / z -\ 2J2.1 7 JL / 4. 2. 5-r. od. 7. ii cwt. 2 qr. 12 Ib. i f \ IIS! \ e l 4:0 ) : \ 58 _5 104 (/) ^f ^ 5. 4 ton 8 cwt. 8. 26. gs. >]\d. 620 A MODERN ARITHMETIC Exercises XLV. a. P. 163. i. () A. (*) A. (<) if, w Y4 8 o> w imp 2. (a) I, (3) I, (c) J, (flT) i (*) i, (/) w, () f - o / 4 15 8 7 \ / 1 5 7 5 3 \ / 8 60 7 \ / 2 2 \ 3 - (12 45- 2T FT/I (TF so F2V> (TF rso TT/> vw e / \ 144 30 24 48 96 96 . 5 - W 2TF> T5> F> T> T4~4> TT4"? 162 30 24 54 108 96. " " " T^F? TT4 /xl08 30 24 36 72 96 (') -2TF FTF> Tff> TTi T4T' T92"' 6. (a) 21 lb., (b) 5 lb., (r) 21 Ib. 7. (a) 8 in., (3) 9 in. 8. 24. 9. T 2 T and 4 8. 1. Exercises XLV. b. P. 164. If, (*) }ft W f i, (rf) , W II, (/) if 8, (^) JTT, Exercises XLVI. a. P. 167. 1. 3 cwt. 16 lb., 2 cwt. 91 lb., 5 cwt. 107 lb. 2. ^8. 05-. 8d., 3. (a) 6||, (3) 7 |f, (r) iiJJ, (rf) 19 J, () 10/2, (/) /A, (0 2 3 Jf, ( A 8 7 7 1 c 5 1 17 79 n 61 11 23 283 4- T7 T8-J Tfff T- 5 - TJ TF> T2"FJ ^TF- 6 - ^6T> T"2 J> TO' TFT' H ^175 ^11 1 x- / 7 7 Q 13 a S3 in r 13 7- 5T92-> 3-52V, 22-53-, 696704- 8. TTT' 9 - TTT' 10 - 'To- ll. ^"6. u. i^. 12. 2. 55. uJflT. 13. I3tonsi9qwt. 55 lb. 14. 2 mis. 7 fur. 195 yds. 15. 30 yds. I in. Exercises XLVII. a. P. 171.. 1. () f , (6) A, to inhr, W I, () I, (/) A, (^) i (*) (0 4 2 o 7 !T, (/) I 8 A, (*) I, (/) 6To, (w) 7 (y) T 2 oV, (r) if f J, (s) 2. () f , T 3 4 5 Ai TT, (*) I, H, li 1% <o fit, if, w i *i, i A. Ji, w T 4 5, A, f , T 2 A- Exercises XLVII. b. P. 172. 2. 3-66666, 3-63, 3-625, S^SSS- 3. S JJ. 4. i^fe. 5. i T 4 ^ ANSWERS 621 Exercises XLVIII. a. P. 174. 2. (a) 6}, () si (c) 7 J, (</) S J, (*)6j, (/) 6, (*-) 3 J (A). S } (0 4lV (/) Si- 3. (a) f, (J) |. 4. () ij, (3) |. 5. I T V 6. T V 7. (a) -T5, (*) A (^ A* M A- Exercises XLVIII. b. P. 175. ! 2 /4- 2. (a) 4 JJ, (3) if, (V) 21. 3. (a) &, (J) &, (<) TOFi W T^ W iVV Exercises XLIX. a. P. 176. 1. (a) f f , (J) iV W Jf, (d) ff, W J^ (/) A. (^) 64 (A) 3. 28 sq. cm., Exercises XLIX. b. P. 177. 1. () irn (*) A (<) Jt W II- 2. (a) ^, (^) iij, (^) i8i (of) |f. 3- (a) 41, (J) T 7 o (0 I4T- 4. (a) SxVo, (*) 2! h- 5. (a) 240, (3) 3 T V 6. 5 2 /3-. Exercises XLIX. c. P. 177. 2. (a) 10, (b} 7, ( C ) 6, (d) 9, (,) 4, (/) 4, (^) 6, (A) 6, (i) 10, (7 o 12 4 32 c /,104 104 /z\fifi fifi c /vlS /z\ 40 o. -7-. *. j-g-. 0. (a) -js"-, -8~o~> (^) TO"' TT' b< ( a ) T3^' W F' . Exercises XLIX. d. P. 178. 1. () 4 81b., 0)7olb., (.)|. 2. |. 3. |f. 4. 24. 5. f}. 6. 2. 7. 128. 8. 9. 9. gV 10 - 22^. 11. 20. 12. 7. Exercises L. P. 180. 1. '(a) |, (6) 3?. W ii (rf) w W -rfi 2. (a) |, (3) i (r) f ftr or 3. (a) f , (*) - 8 g 5 -, (<:) 4. (a) f , (*) T 4 /2 , A MODERN ARITHMETIC Exercises LI. P. 180. 1. A- 2. T V 3. fi 4. #. 5. T V 6. TF- 7 - TlJT- 8 - T2- 9- J>TTf 10. 907 tons 4 cwt. 11. -W-, 2 ff. 12. f. 13. iff. 14. J. 15. f 16. I T V 17. (b)(c)(a)(d). 18. t^V 19. (a) 5!, () if? (<:) 2f. 20. () 3^1, (^) ijf- 21. (a) ifff, (^) n. 22. (a) 3, (^) I3lff. 13 11 >9 21 47 143 181 23 2fif557 ,,179 23. 32^(5", 2 9B"' 2 260") 23-2, Sy-gl', 3T^2J 3l9"^' 3^ 3"T47^" 324T7* 24. (a) T 8 Ai (*) Hf 25. (a) 26. (a) fS, (*) III, w fit- Exercises LII. a. P. 184. 1. (a) 5 2 x3 2 , (3) 2 2 x3xii, (c) iix3 2 x2 3 , (rf) S 3 xii 5 W 3 2 x2 3 x7, (/) 2^, (^) 2, (h) 3X2^, (0 5 X 2 6. 2. (a) T 4 T , (*) 2T' W TT> W T () f I (/) I' (^) 41' W If (0 TT (/) lil' (^) TTTT (0 I. (') f () TV> (^) A' (/) if ' Exercises LII. b. P. 185. 1. (a) 36, (6) 2700, (0 37, (;/) 144, () 25, (/) 9, C?0 96. 2. 4:5:6. 3. i, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 7i 73, 79, 83, 89, 97. 4. 5 and 4536. Exercises LII. c. P. 187. 1. (a) 1-3 ft., (b) 17, (c) 1-4. 2. 1-5 ft. 3. 6mm., 19 and 14. 4. () 3, (*) 4, (^ 5- 5. 713. Exercises LII. d. P. 189. 1. 6 in., 37. 2. 15*. 3. (a) 131, () 131, (c) 233, (a?) 109, (e) 25. 4. (a) H> (b} 1, (0 T |, (rf) iff, () IJf, (/) Jf, (g) Jf ," (/i) T 8 VI (0 M> (/) t W il (/) i*Ai () f i () T 4 2T> () IT. (/) TrV 5. 6 c.c. 6. 429 wts. of 12. 7. A cube of 1-5 in. edge. 8. 8944. 9. 7-6 ft. 10. 326 and 49 loads. 11. 94 in. 12. 32. 13. 24 Ib. 14. 96 soldiers, 49 rows. 15. 6s. $d. 16. 63 balls, 90 groups. ANSWERS 623 Exercises LIII. a. P. 192. 1. 3 2 x2 2 . 2, 3 5 x5 4 . 3. 2 4 x5 3 x3 2 . 4. ii 2 x2 4 x 5 3 x3 3 . 5. 7 4 x37x5 4 x2 3 . 6. I3 2 x2 5 . 7. 2 10 x3 4 . 8. 5 3 x 2 6 . 9. 2 5 x 7 1 x 3 2 . 10. 3 4 x 5 x 2 4 . Exercises LIII. b. P. 193. 1. 3 4 x 5 3 x 2 4 x 7. 2. 7 2 x 2 6 x 3 3 x 5 x 1 1, 4656960. 3. (a) 390, (6) 19200, (f) 444, (of) 504. 4, 3 20 15 2 4 . / x 2 7 6 288 9O 280. / \ 385 255 l) 6 F' 6"0> 15TF 6~0 ) () 6 CTOJ 6"0 0> ffTFTF* 6" 00 > M T3 6T> 1 3 6"5> 390 819. ,j\ 420 f > 5 1 1309 48 . ,\ 864 " ) TT^8"> 2 B T4 SF 832 702 143 T8T2~> TT^ T8T2"' 209 125 62 . /7 x 10 32 23 17 . / x 59 85 109 2~> " " " " ^ T8 (TJ F> T~9 J > ") 2~> TT> T6> T * "5 4 > T8"' "9 3 6"' 6. ios. 7. 12 ft. 8. 108 c.c. 9. 4 sq. ft., 16 and 9. 10. 12 sq. ft., 64, 27, 12. 11. oVV 12. fgV 13. ff. 14. T 8 A- 15. 2%. is. -Hi- 17. A 9 ^- is. 2M^ 19. 204. 20. 19. 21. At every of an inch. 22. 3026. 23. 603. 24. 751. 25. 600. 26. 348 Ib. 27. i. 28. 24. is. od. 29. 1836. 30. 1 120. 31. 93 yds. i ft. 32. 19.5-. and 24 fr. 33. 54. 34. 246. 35. 54 and 72. 36. 224. 37. 0-45 inches. 38. (a) 7, (6) i, (c) 31. 39. 69696. 40. (a) 5184, (6) 93636- 41. (a) 421875, (b) 1728, (c) 42875, (d) 287496. 42. 21472,502. 43. 225. 44. 13, 34, ..., increasing by 21, Exercises LIV. P. 202. L 25 kilo. 2. 770 Ib. 3. 150. 4. 9-9. 5. 60 sec. 6. 66. 7. 95-68 min. 8. 14. 9. 50. 10. 52500 cub. ft. 11. 6s. 3^. 12. 60,000 Ib. 13. j^i. 4j. $%d. 14. 216. 15. 24 yds. 2 ft. 8 in. 16. ^9500. 17. loo- 1 sq. ft. 18. 14080 Ib. 19. 200. 20. ios. lod. 21. 42. 22. 0-0804384 Ib. 23. 309300. 24. 4-09 Ib. 25. 350 ft. 26. 1800. 27. iS. iSs. od. 28. 4. i6s. od. 29. 6s. <)d. 30. 1 200 tons. 31. 174 tons. 32. gV sq. in. 33. 140 ft. 34. 9 cwt. 35. 7 Ib. 15 oz. 36. 63 in. 37. 5500 gal. 38. 94-5 Ib. 39. 8240 Ib. 624 A MODERN ARITHMETIC Exercises LVI. P. 209. 1. 3729 nearly. 2. 17 ft. 3-24 in. 3. 806-4. 4 - 633-024. 5. 1050. 6. 22 ft. per sec. 7. 35-325 ft. per sec. , 540. 8. 30. 9. 69-8. 10. 0-2355 miles. 11. o-ii68cu. ft. 12. 16800 cu. fU 13. i ton 13 cwt. 2-4 qr. 14. 12-21792 tons, i.e. 12 tons 4 cwt. 1-4336 qr, 15. 13300 cu. ft. gas, 2100 Ib. coke, 98 Ib. tar. 16. 2-1525 kilo. 17. 2-46 gms. 18. 0-68 Ib. 19. Passengers 172254, goods 179550. 20. 100, 1375, 825. 21. Sports 30, football 45, cricket 120. 22. 75, 100, 175. 23. 19-2864 in., 20-0592 in., 9-6544 in. 24. 147150 Ib. 8 oz. true to nearest oz. 25. .799. 45-. io|^., 985. 135-. o%d., 640. 2s. id. 26. 221, 184, 77. 27. (a) 1040. i2s. 6d., (b) 1271. ijs. 6d., (c) 1965. 12s. 6d., (d) 3121. ijs. 6d. 28. ji. 5-r. $ld., 1. gs. 6$d. 29. 10-4 tin, 89-6 copper. 30. 2 tons 16-025 cwt. 31. 20. 32. 91-67 gold, 8-33 copper. 33. 11-54 tin, 11-54 zinc, 76-92 copper. 34. 85-47 tin, 14-53 antimony. 35. 39-62 copper, 20-13 sulphur, 40-25 oxygen. 36. 18-18 lime, 54-55 sand, 27-27 water. 37. 74-67 carbonate of lime, 6-00 clay, 19-33 san d : 15 tons 2-4 cwt. 38. 78-84 clay, 0-55 coal dust, 20-60 nitro-glycerine, 6-633 kilo. 39- 57'5 sodium, 40 oxygen, 2-5 hydrogen. 40. i ton 16-4 cwt. 41. 21. i6s. &d. 42. 41. \y. 4 d. 43. yV> J, T, |. 44. 25, i2i, 2j, 2, 75, i6. 45. (b) 16-33, (c) 2-202. 46. (a) 0-63, (b) 102, (c) 0-49, (d) 0-30. 47. 1-862 tons. 48 - 6g %, 2. Ss. i id. nearly. 49. 4*. Jd. 50. IOT. 51. is. 6d. 52. 5. 53. 3. 54. 27-2 loss. 55. 21. 5*. od. 56. ^19. BJ. 4d. 57. 2-5. 58. 4 ^f . 59. 192. los. od. 60. 45 to 35. 61. 2fc. 62. 0-0153. 63. i sq. ml. 207 ac. 6 sq. ch. 64. 9900. 65. 2-34375. Exercises LVII. b. P. 218. 1. 600 yds. 2. 55 yds. 3. 28-5 ft. 4. 7-29375 mis. 5. 1-305 mis. 6. 19-23 yds. 7. 66-332 yds. 8. 6-524 metres. 9. 83-64 cm. 10. 10-53 ft. ANSWERS 625 Exercises LVIII. P. 222. 1. (a) 6 ft. 2 in., () 14 ft. 2 in., (c) 18 ft. 8 in., (d) 31 ft. 3 in., (e) 14 ft. 2 in., (/) 15 ft. 3 in., (g) 21 ft. 2 in. 3. (a) 45 miles, (b) 53 miles, (<r) 2 miles. 4. (a) 107 miles, (b) 37 miles, (<r) 94 miles, (^) 103 miles, (e) 97 miles, (/) 117 miles, (g) 63 miles. 5. (a) 22 miles, () 26 miles. Exercises LIX. P. 224. 1. i, 2, 3, 5 7, ", 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59; 3 15 . 2. 503. 3. ^1774- IS'- i<* 4 - ;5- 4*.6<*; 0-25. 5. A 2. i2j. 6rfl, B 4. i6s. 3</., C 4. icw. orfl 6. 0-20432 ; 3 2-ijo o- 7 - IIS- 8 - 1262250. 9. sJ Ib. 10. 252 Ib. sulphur, 378 Ib. charcoal, 1890 Ib. nitre. 11. T f -g" 12. (a) 201%, (*) I ton 15 cwt. 3 qr. t (0 3200. 13. 32. 14. ii. 15. 4 tons o cwt. 3 qr. 16. 6. 6s. od. 17. 2 4 x 3 2 x 1 1 2 ; 6 chains or 0-075 mile - 18 - l- 4 s - 2 $d- 19. ii. 20. ^390; 4 J. 21. i. iSs. &d. 22. (a) I T V, (6) 0-2. 23. (a) , (3) I A- 24. if days. 25. ^75. 26. 45. 27. 2025. 28. 425. 29. 56718-75. 30. 64. 31. 8. 32. 34. 33. ii, 13. 34. 8. 35. A wins by y mile. 36. ^212. los. od. 37. ^"o. 125. o^d. 38. 0. i6s. \\d,-, J. 39. A 6. 5^. orfl; B 6. 15^. o^.; C 7. los. od. 40. i8|. 41. 39 4!, 42. 5. ifo. 6$d. 43. 165. 44. ^5. 45. 25. 46. 3,7,"- 47. 0-00288580. 48. 4 cwt. copper, 10 cwt. zinc. 49. ^2000. 50. ^78. 15*. od. 51. 7 hrs. 52. 0. 9*. 4^; ^"o. 5^. 7^.; ^"o. is. lojdl; 2^., i^., i^. 53. 3i. 54. ^385. 55. A ^300, B ^98, C 112, D 120. 56. 792. 57. 0-0014999. 58. 7. 59. 1671 oz. 61. A 540, B675, C^i20o. 62. 3%. 63. 200:243. 64. 45. 65. 526. 66. m- 67. 4 flf f I- 68. }JJ, Mf 69. 82/4. 70. ^83000. 71. ^33- 6s- &/. 72. ^"7173. 6^. Sd.', ^8070; ^8966. 135-. 4^.; 8608. 73. ^60. 15^. od. 74. 151. is. Sd. 75. ^"2418. 17s. &d. 76. 140, 135, 126. 77. T V 78. 0-635- 79- 15- 80- 25. 81. 2. gs. &d. 82. 0-20625, 4. $s. 9 -6d. 83. (a) JJ (*) iV, (0 25! 84 - 52O. J.M.A. 2R 626 A MODERN ARITHMETIC 85. (a) 189 miles, (3) 28 miles per hour, (c) 10.30 a.m., 5.15 p.m. 86. 328. 4s. 4\d. 87. T V 88. 3. IQT. od. 89. man 8. i&. i\d.; woman 7. 2s. 6d.; boy s- 6s. io\d. 90. o. is. 4\d. 91. 3. 92. 603. 5*. od. 93. A 17309- 5 J - orfl, B 8654. i2s. 6d., C 6923. i 4 j. oaT., D 5769- 15-r- <** 94. 2160. 95. 480, 15 of 32, 10 of 48, 12 of 40. 96. 10. 17-r. ii$d. 9T. 10. ior. yd. 98. A 90, B 81, C 108. 99. irVV 10 . 5x3 2 xn, 3x11x13, 5x3x13, 6435. 101. 306. i6s. od. profit. 102. 76. i;j. id. 103. i. 6s. Sd.; f. 104. 45 weeks. 105. 14. 106. 904. 107. 72^?i- 108. 1240, 806. 109. 74088. 110. 25-25. 111. iof. 112. 479. iij-. lod. 113. 108909. 114. |. 115. 50000, 6000, 6600, 3740. 116. io T 4 T 7 (r- 117. 113-0016 grs. 118. A pays B 2s. yd. t C pays B 3^. 2d. 119. A 13. ior. od. t B 14. 8j. od., C 14. 145- od. 12 - ^35 per horse, 14 per cow. 121. A i. 45. od., B o. iSs. od. 122. 1234. 4j. od., 617. 25. od., 205. 145. od. 123. k ,202. los.od., 6^225, C;ii2. 105. od., D 75. 124. 56250. 125. 34 |. 126. o. los. ud. 127. o. 45-. ofaf.; 0-05078125. 128. 117. 14^-. ud. 129. 12. 130. 683. i 3 j. oidl 131. 3-25. 132. 330. Exercises LX. a. P. 237. 1. (a) 35. 6d., (b) 35. iod., (c) 3 s. 7 d. t (d) 45. ted. 2. (a) 7s. Sd., (b) 45-. 2\d., (c) 6s. iod., (d) Ss. 2\d. 3. (a) 45. gd., (b) 3j. 1 1 JaT., (^ 5*. n\d., (d) us. $d. 4. (a) 125. 6d. t (b) 145. ftd., (c) iSs. Sd. 5. i. is. i\d. 1. 5. 9. 13. 17. 22. 26. 4. 7 s - >\d> 3*- 6Jdl 69 qr. $. 15. Ud. 3. 125. lid. 513. \\d. in the . Exercises LX. b. 2. 107. 7^. 2d. 3. 6. 1578 Ib. 7. 10. 26600. 11. 14. g. igs. Sd. 15. 18. S. Ss. sd. 19. 23. 636. 24. 27. 8 T 2 T miles. 28. P. 237. ior. 2d. 4. sj. i$dl 380 cwt. 8. loo qr. 12100 tons. 12. 12200 tons. 2. IQS. 7d. 16. 4. iss. 4d. 7. 20. 1126000. 21. 24. 521000. 25. 2179. 3 tons 2 qr. ANSWERS 627 1. 26-060. 4. 35. 2s. 2d. 7. 83. I3s. nd. Exercises LXI. a. 2. 26-073- 5. 98-922. 8. 34. 2s. 2%d. P. 242. 3. 35- 2*. 6. 15-583- 9. 17. i 7J . Exercises LXI. b. P. 242. 1. 2-1. 2. 2-15- 5- 3-85- 6. 4-75. 9. 0-00 1. 10. 0-003. 13. 0-024. 14. .2-034. 17. 12-431. 18. 12-481. 21. 13-106. 22. 13-365- 3. 5-25- 7. 102-65. 11. 0-005. 15. 12-041. 19. 12-091. 23. 14-305- 4. 5-4- 8. 116-95. 12. 0-009. 16. ii-534. 20. 101-169. 24. 114-255. Exercises LXI. c. P. 242. 1. 2-675- 2. 3-419- 3. 101. 2s. $\d. 4. 6-851. 5. 3-75i. 6. 7-804. 7. 3-705- 8- 3- 5*- 9^. 9. 6. 2s. Q\d. 10. 12. 6s. o^d. 11. 8. 5*. lojrfl 12. 16-929. 13. 3-817. 14. i. i4j. 6|ar. 15. 2. 6s. &d. 16. 100-100. 17. 100-092. 18. 23-896. 19. 15-336- 20. 3. y. tf. 21. 2. 2s. 8^. 22. 2-177. 23. 7-059. 24. 13. os. \\d. 25. 8. 2s. o^d. 26. 9-131. Exercises LXII. P. 244. 1. (a) (i) 186. I3J. 4 rfl, (ii) 1866. iy. tf., (iii) i 4 93- (3) (i) i. iu. 2|^., (ii) o. 4J. sirfl, (iii) IQJ. sdl ; (^) (i) 751928. 2s. 6d., (ii) 361503. i&. iK 2. 150- 3. (a) 438- iw. io^., (*) 9". 17-r. 6^., (f) 155. 4*- (rf) 1582. i 9 j. 4i*. 4. (a) o. 4^. 2d., (b) 2. 6j. &/., (c) 21, (</) 8. /., (iv) 896; Exercises LXIII. a. P. 246. 111111 1111 i ' > T T> T^ TO"' ~5i > T5~> "2 Ollllll * TF> 3 T ^J ? T2"' 4 > } J T' T> jV" i i "2 4> T 628 A MODERN ARITHMETIC Exercises LXIII. b. P. 247. 1. () (*) T> (') TV' (d) T5> (*) T<T> (J 2. (a) J> (3) i, (0 i> (<0 J () irV (/) TO 3. (a) i (J) i (<0 J, (rf) T V> () TO (/) sV 5. (a) 9-r., (b) I3s. 6d., (t) iSs. 6. (a) gs. 4d., 7. (a) 2, (6) 4- ijr. 4^- (') ^5- 6.r. &/. 8. (a) 2f. 6^., () 2s. id., (c) is. Sd. 9. (a) i, (*) 2, (c) 2s. is. 4d., (d) i. iSs. Sd. 10. (a) i, (J) 3, (r) 4, (d) 4. 3*- 4<*. (*) 3- *&. Sd. 11. (a) I2s., (b) 9^., (f) 5. us. od., (d) i6s., (e) i6s. 4d. 12. (a) i, (b) 2, (c) 2. is. Sd., (d) i. iSs. 4 d., (e) 4. os. iod., (/) 5. os. iod. 13. (a) i, (b) 2, (c) 2. os. 7^d., (d) 3, (e) 3. 2s. 6d., (/) 5. iSs. gd. 14. (a) i, (b) 2, (c) 2. y. 4d., (d) 4, (e) 54. y. 4 d., (/) 200. 15. (a) i, (*) 9, (e) 2, (d) 16. i;j. firfl, (e) 54, (/) 109. w. 6d. Exercises LXIII. c. P. 248. 1. (a) 16. Ss. iod., (b) 24. ly. 3d., (c) 32. ijs. Sd., (d) 49. 6s. 6d., W65- 15^4^., (f) S2.4s. 2d. 2. (a) 14. i6s. \o\d., (b) 27. 45. 3%d., (c) 32. 3s. 2$d., (d) 47. os. id., (e) 54- 8j. 6|^., (/) 71. 145-. io%d., (g) 86. iu. g%d., (h) 66. iss. ii^d., (i) 89. is. 3d., (j) 103. i8j. \\d. 3. (a) 65. 145-. od., (b) 99. 155-. 4d., (c) 136. y. 4^., (d) 160. I2s. od., (e) i99. IO.T. Sd., (/) 214. 2s. Sd., (g) 255. ioj. od. t (h) 282. 5j. 4d., (i) 311. 9j. 4^. 4. (a) 457. 25-. ejrfl, (b) 2545. 3j. gfrfl, ( f ) 2936. gs. 2\d., (d) 3428. gs. ofrfl, () 3486. nj. 3^., (/) 3409. u. 8^., (f) 35 6 4- or. lOflT., (A) 3602. iss. 5. (a) 11325- ios. Sd., (b) 18938. 9 j. gd., (c) 23337. 45. (d) 101763. I3s. $%d. 6. () i7i57. iy. ii^. (*) 25460. i;j. 2^., ( f ) 31444. 4J. (^) 39605. or. sf at 7. (a) 107017. 9*. 7fc*, (3) 160415. of. 3|flT., ( f ) 113615. 14*. (d) 268470. 8f. sJaT. ANSWERS 629 8. (a) 69659. 12s. 6d. t (b) 80558. i$s. orf, (c) 61666. i&. 8</., (rf) 41069. 35. 4^., () 151403- u. 3<* 9- (<*) 390813. 2J. 8irf., (3) 151702. igs. 5i<, (0 292019. 91. 7|<, (flf) 409601. l8j. 2fl?. 10. (a) 21518. ifc. 3rf, (6) 35864. 13*- 9^, (0 549o8. 2s. Set., (d) 45006. 13*. 4d. t (e) 19409- 2J. 6d. Exercises LXIII. d. P. 249. 1. {) 50. 13*- 7i*, (*) 68. iu. 4i</., (c) 101. yj. 3^., (aT) 152. or. ioK, () 281. 14^. 6iflT., (/) 324. i% iJaT. 2. (a) 136. 5 J - i^. (*) 343. 19^ 4^-, (0 599- 3^- i^, (rf) 2271. i3j. liar. 3. (a) 1292. iu. 7^., (^) 583. i2j. 9rfl, (r) 2238. i8f. 8A, 4. (a) 14. ;j. cwT., (6) 144. I9f. 6^., (r) 571. 7s. 2\d., (d} 873. I4J. loaT. 5. (a) 141. i&. 4flT., (*) 2890. sj. 6ct., (c) 4008. isj. 8^., (d) 6151. i8j. iaT. 6. (a) 455- 9^- 2^., (*) 1300. I2J. iia?., (r) 3358. 15^- 9^, (^) 5771. 2j. 6d. 7. () 97- 14-f- 4^-, (*) !22. SJ. iiaT., (0 596. Of. laT., (rf) 981. 8. (a) 1147- i8f. 4^-, (*) 5939- I0f. &/., (c) 6877. i 5 j. 7fl r., (rf) 12519. 135-. gd. 9. (a) 1512. I3J. oar., (^) 8291. i&. SaT., (r) 15756- 7^- ^, (^) 7324- lor. 6rfl 10. 142. 8j. loaT., (*) 214. 7s. qL t (c} 1710. 4*. orfl, (^) 3964- I5-5-- "^. Exercises LXIII. e. P. 249. 1. (a) 2J. 6^., (b} is. Sd, (c) gs. 6d. 2. (a) I4s. 3^., (d) 40*. 6flT., (r) 17. ioj. 6rfl 3. (a) 3^, (*) 4^. (0 2J. 4. (a) ud., (d) &d. t (c) \\d. 5. .(a) i. I7J. 6d. t (6) 5. 17*. 6d., (c) 36. 6. (a) 2^f. t (b) 4-r. loiflf., (r) 2rf. 7. (a) 11. 12*. 9dT., (^) 6|rfl, (r) 7j. 6^1 8. (a) S rf., (*) ioK, (0 22s. 6d. 9. (a) 2^., (b) 5*, (r) &. i^. 10. (a) i Jar., (<5) i. 2s. 6d. 630 A MODERN ARITHMETIC Exercises LXIII. f. P. 250. 1. (a) 26. i6s. 6d., (b) 66. os. &/., (c) 123. i6s. tf. 2. (a) 240. gs. od., (5) 330. I2s. 4%d., (c) 490. 185-. t&d. 3. (a) 93- 14*. "i^M (b) ;354- i&. 2*., (*) i499- 19*- &/. 4. (a) 496. iSs. S^d., (-5) 851. i7j. grf., (t) 3738. i6j. loet. 5. () i. I2J. S$d., (*) 2. i8j. loirf., (<:) 4. 5*. o^. 6. (a) 231. igs. i\d., (b) ^651. iu. 3^., (f) ^1652. 15*. 7. (a) ^475- "J. ioK (^) ^2055. I2J. 5irf. 8. (a) ^131. &. iifrf., (*) ^120. IOT. sfaT. 9. () ^"402. 3j. &/., (b) ^848. 145-. 6c?. 10. (a) 3. igs. 7rfl, (^) 6. I4J. 3l^. 11. () ^218. los. &., (b) 327. iss. Stf. 12. (a) us. 6%d. y (b) i. sj. 5J^. 13. (a) 302. 2s. 4jrfl, () 402. i6j. 6^. 14. (a) 6. 2s. 8^., (3) 2. 7 s. &d. 15. (a) ;86o. ijs. id., (b) 227. Ss. 6\d. 16. (a) 4. i7s. Sd., (b) 19- 6s. gd. 17. (a) 2. 4j. &/., (^) ^8. or. 2rfl 18. (a) 24. 5^. 4jrfl, (b) los. G\d. 19. (a) ^144. 13^. od., (b) 11. los. od. Exercises LXIV. P. 251. 1. .109. 45-. 4\d. 2. ^586. 3j. 2d. 3. 16 tons I cwt. 3 qr. 4. 5. 211. 2s. S^d. 6. ^10850. ios. gd. 7. ^33. 2s. 7 T V- 8. ^162. is. io^d. 9. 21. gs. 4 10. 796,603-5 tons. 11. 7. 14$-. 6^. 12. 281 mi. 978! yds. 13. 5 cwt. 25-9375 Ib. 14. 3. 2s. 4%d. 15. ^"1524. us. 2\d. approx. 16. 270. 17. 7- 17*. 2^d. 18. 57 ton 4 cwt. 2 qr. 7-^- Ib. 19. /i6. 7j. 20- () ^31- 5 s - 4^- (*) ^"1017. i4j. &/., (r) ^11284. I4J. o (^) ^557- u. &f., () ^10609. ii^. orfl, (/) ^2891. Ss. od., (g) ^8916. is. od. (K) ^6804. i8j. Sd., (i) ^1286. iu. 4^. (/) ^65083. 4j. o^. (^) 1234. or. orfl 21. (a) ^2138. iss. Sd., (6) 2iS3- 5*- orfl, (0 ^20944. 22. (a) 15, ^435, (3) ^520, 500, (c) 700, 668. ios. od., (<0 ^75, ^1425, (^) 800, ^40, (/) ^350, g. ios. od. ANSWERS 631 23. (a) 13. 15*. iod., 24. (<?) 56. 17-r (/) "633. (m) 220. 53 (a) 400, (e) 5000, (b) 741. os. iod., 35-. 2d., (e) 48583. 4-r. iod., Id., (A) 228. igs. 6d., l$s. ^d. (k) 12. 12s. $d., 2d., (n) 255180. 1 6s. od. (b) 6800, (c) 5600, (/) 8125, (g) 17000, (C) 109. 2S. 2d., (/) lO. 12S. I0rf., () 45060. I or. 6d., (/) 76. 17*. it*/., (rf) 78000, (A) 560000. Exercises LXV. a. P. 256. 0-5 sq. in. approx. 3. 0-95 sq. cm. 4. o sq. in. approx. Exercises LXV. b. 1. 1-24 sq. in. 2. 3. 0-55 litre. 4. 5. (a) 1644-3025 and 1636-2025 sq. in. (b) (c) 355-3225 and 35I-5625 sq. in., (d) (e) 642-6225 and 637-5625 sq. in., (f) 6. 1128-7, i47- 7. (a) 1 1463 and 10938-6, (b) (c) 1 5000-0 and 14415-8, (d) (e} 2299-5 and 2245-5, (/) 8. 0-22 and 0-23. 9. (a) 5-88 and 6-00, (b) 4-96 and 5-04, (d) 1-87 and 1-88, (e) 2-44 and 2-54, 10. 95778 cm. and 94757 cm. 11. P. 256. 648-48 cub. in., 625-33 cub. in. 0-0183 ac - 126-5625 and 124-3225 sq. in., 583-2225 and 578-4025 sq. in., 3606-0025 and 3594-0025 sq. in, 6944-6 and 6596-8, 47388-2 and 46099-9, 32801-0 and 32459-0. (c) 6-58 and 6-68, (/) 2-49 and 2-57. 18-4 and 18-1. 1. 97000. 2. 21 X IO 4 . 3. 5. 1-6 xio 5 . 6. 4-6 x io 6 . 7. 9. 4-4 -f i o 3 . 10. 1-8 xio 6 . 11. 13. 96 xio 1 . 14. 4-0 x io 6 . 15. 17. 9-4 xio 4 . 18. 3-6 xio 3 . 19. 21. 4-5 x io 1 . 22. i -i x io 5 . 23. 25. 2-9 x io 3 . Exercises LXVI. a. P. 264. 336 x io 3 . 4. 27 x io 2 . 4-2 x io 3 . 8. 2-3 x io 6 . 30 X IO C . 12. 2-2 X IO 2 . i-oxio 3 . 16. 3-0 x io 7 . 1-3 x i o 6 . 20. i-8-rio 1 . i-Sxio 1 . 24. 6-0 xio 3 . 6 3 2 A MODERN ARITHMETIC Exercises LXVI. b. P. 265. 1. 4724 x io 4 gr. 2. (a) 131 x io 3 , 72 x io 3 , (b) 2-3 x io 3 , i -i x io 3 , (c] 3-50 x io 3 , 1-91 x io 3 , (d) 1-69 x io 4 , 9-5 x io 3 , (e) 3-3 x io 3 , 2-0 x io 3 , (/) 4-7 x io 3 , 3-0 x io 3 , () 2-ox io 3 , i-i x io 3 , (k) i-ii x io 3 , 7-6 x io 2 , (i) 6-6 x io 3 , 4-6 x io 3 , (/) 3'0 x io 3 , 1-8 x io 3 , (k) 9-3 x io 3 , 4-9 x io 3 , (/) 2-7 x io 3 , 1-9 x io 3 , (;) 5-3 x io 3 . 2-4 x io 3 . 3. 88 cub. metres. 4. 24-40 ft. 5. 57560, reliance on ist three figures. 6. 3-50 x io 4 tons. 7. Elevation in feet o 328 656 984 1312 1640 Bar height in inches 29-92 29-57 29-21 28-85 28-50 28-19 8. 144-8. 9. 2. 10. 60000. 11. 390655. 12. 327235-8. 13. 61879-2. 16. 835216-8. 17. 107-98. 20. 9-01. 21. 18018. 24. 28763 xio 3 . 25. 3619 xio 3 . 28. 65771 xio 2 . 29. 403 xio 4 . 14. 29629. 15. 17-5364. 18. 981598. 19. 1-0612. 22. 9994-5. 23. 4229. 26. 1 79442 xio 2 . 27. 1 827 xio 3 . 30. 8831 x io 4 . Exercises LXVII. P. 271. 1. 6-458. 2. 3869. 3. 13 lb. 4. (a) 60-8233, 2 7' 0-19". 5. i : 2566, I : 1841, I : 1609. 6. (0)4287, (b) 5202, (c) 626, (d) 554, (e) 1161. 7. 561-24 ft. 8. (a) 999-41, (b) 31-265, (c) 25768, (d) 0-81688, (e) 208-01 x IO 3 , (/) 106-54. 9. 7 -680 in. 10. i-6o9kilom. Exercises LXVIII. a. P. 273. 1. i, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400. 2. o, i, 4, 5, 6, 9. Exercises LXVIII. b. P. 273. 1- 42. 2. 210. 3. 75. 4. 231. 5. 84. 6. 36. 7. 48. 8. 1540. 9. 54. 10. 63. 11. 210. 12. 66. 13. 256. 14. 1155. 15. 192. ANSWERS 633 Exercises LXVIII. c. P. 273. 1. (a) 2, (6) 3, (c) 6, (d) 10, (e) 9, (/) 14. 2. (a) 2, (b) 4, (c) 12, (a?) 8, (*) 8, (/) 10. 3. (a) 3, (6) 7, W 14, (d) 14, () 13- 4. (a) I, (3) 2, (0 2, (aT) 2, (*) i, (/) 2, () 3, (A) 3. 5. (a) I, (J) 2, (0 2, (aT) 3. 6. (a) i and I, (b) i and 2, (c) i and 2, (a?) 2 and i, (e) 2 and 2. Exercises LXVIII. d. P. 277. 1. (a) 13, (6) 17, (c) 23, (</) 29, (*) 47, (/) 53. 2. (a.) 58, (6) 64, (f) 74, (d) 85, (*) 89, (/) 93, te) 97, W 99- 3. (a) 504 yds., (*) 1012 yds., (c) 1228 metres, (d) 1752 ft., (<?) 2396 yds. 4. (a) 672, (3) 709, (c) 856, (a?) 945, (e) 2041, (/) 3872, (^) 4018, (h) 5306, () 5084, (/) 70963- 5. (a) 23-04, (b) 70-082, (*) 0-924, (a 7 ) 491-62, (e) 6-57, (/) 30-0905, (g) 0-0529, (A) 0-02538, (0 0-008904, (/) 64-007. 6. 29, 23 over. 7. 278 yds. 8. 68-08 sq. cm., 219-04 sq. cm. 9. 10 ft. 7 in. nearly. 10. 182 ft., 730 : 728. 11. 19 people, iqd. Exercises LXVIII. e. P. 278. 3 2 o!98 yi9 IT' 3 - 2TT- 4 - 44 6. |fj. 7. 1-59. 8. 0-77. 9. 0-86. o 3 2 o!98 yi93 e219 2. IT' 3 - 2TT- 4 - 446-' 5 - TT98- Exercises LXIX. a. P. 280. 1. 46. i6s. gd. 2. ^27. I4J. i^. 3. 32. Exercises LXIX. b. P. 282. 2. ^38. icw. 40 7 . 3. ^40. 19^. 70 7 . Exercises LXIX. c. P. 282. 1. i. 5-r. od. 2. \d. 3. ^o. u. 8a. 4. 0. fo. 3flf. 5. ^o. 6s. Ba. 6. ^o. 3*. 40. 7. 40. 8. 1. 9. /,i68. 10. ^42. 11. ^36. 12. ^120. 13. i. 14. 9. 15. 6. 16. 12. 17. 4. 18. 9. 6 3 4 A MODERN ARITHMETIC Exercises LXIX. d. P. 285. 1- ;i3i- S s - 0^- 2. 34. i2s. sd. 3. 356. i8j-. gd. 4. ig. 5.5-. 6rf. 5. 86. 14*. 10?. 6. 153. 13*. 9*?. 7. 16. 15*. 60?. 8. 836. 3 j. 3d! 9. 3. 6j. yd. 10. 54. 17.$-. i od. 11. 19. \2s. 4</. 12. 44. 13. 571. 6s. 2d. 14. (a) i. gs. 7 d., (6) 5. 6s. jtf. 15. 13. i2s. iod. 16. 170. oj. $d. 17. 1012. 95-. yd. 18. j. 19. 2 yrs. 20. 3 yrs. 21. 5 yrs. 22. 5 yrs. 23. 5 yrs. 24. 2j yrs. 25. 4 yrs. 26. (a) 2. js. ^d., (6) 4. los. nd., (c)6i. 125. lid., (d] 108. 14^. na?. 27. 4 yrs. 28. 15 yrs. 29. 30 yrs. 30. 0-2 yrs. 31. 24. 32. 2. ' 33. 31. 34. 2 |. 35. 3. 36. 24. 37. 24. 38. 2 |. 39. 44. 40. 60. 41. 2. 42. 4. 43. 159. y. 4a?. 44. ^394. 4^. oo 7 . 45. ^439. &. 6^1 46. ^575. 47. ^2700. 48. ^240. 49. ^312. 50 - ^1053. 6s. Sd. 51. 318. 15^. oaT. 52. ^150. 53. (a) 30, (d) 60, ( f ) 26, (aT) 26|, (e) 15-3, (/) 30. 54. (a) 25, (6) 15. Revision Papers. P. 289. A 1. 68 tons 8 cwt. 2. i. Ss. io$d. 3. 3704. 4. 1 1 yds. o ft. 4 in. 5. if. 6. 6J. 7. 7-55. 8. ^1-624. 9. 4-191 sq. ft. 10. 68. 11. TfV 12. 0-0875. 13 - 2. 14. 5. 15. o. 4j. lodl 16. 587 ft. 6 in. B 1. f - 2. 42 sq. yds. 3. 3388. 4. 0-0852, 0-0075. 5. 0-0353073, 0-00161. 6. 28. u. od. 7. 16380. 8. 455- 9- 0-13. 10. 760 Ib. C 1. 1167 ft. 3. 5 tons 13 cwt. 2 qr. 9 Ib. 13 oz., 5916264 in. 4. 565, 14 Ib. 5. o. 4j. 2d. gain. 6. ^5 gain. 7. 10%. 8. 180. 9. 10. $s. od. D 1. 9 hrs. 2. ^87. ioj. orfl 3. 20 min. 50 sec. 4. 85 %. 5. 700 sq. ft. 6. 441. 7. ji 4 . IQJ. 740?. 8. 3 J%. 9. 5||. 10. i. 2s. 6d. 11. 34%. E 1. |. 2. 1562-5. 3. ^531. i 3 s. tf. 4. 31 Ib. 4 oz. i6dwt. 16 gr. 5. 0-9416. 6. ^1092. 15^.8^. 7. II5S- 8 - 7- I5-5-- otL 9. 25. 13*. tf. ANSWERS 635 F 1. 670. Ss. id. to nearest id. 2. 7500, 0-0051. 3. 437. 4. T 5 . 5. T V 6. 14-553. 7. 688. IOT. 8rfl 9. 70. 10. 20. 11. i. 5s. od. 12. 7 : 5. 13. 33-3;^. 14. 600, 594, 609, 621, 596, ^646. G 1. 0-697. 2. 1295 litres, 23 litres. 3. 8-279... %, 690900 approx. 4. Manitoba 25, Ontario 17. 5. ,19. II.T. iod., 109. us. lod. 6. 1-85 sq. in., 73 Ib. approx. H 1. ^680,^510,^340. 3. 70% town, 30% country. 4. 37. 5. 21 min. 6. 12800. 7. 1256-636, 4-53 gal. 8. 43-93 yds. 9. 1650 sq. ft. 10. 360. 636 A MODERN ARITHMETIC PART II Exercises LXXII. P. 300- 1. 374 times. 2. 0-1325 mm. 3. 1617-1 yd. 4. 8-75 ft. 5. 0-5 mm. 6. 0-125 in - 7. 5-84 turns approx. 8. 0-14 in., 0-8736 in. 9. 593 cm. approx. 10. 0-05 in. 11. Thread 0-04 in., handle 1-5 in. nearly. 12. 2-632 mm. 13. 1-0094 m. 14. ^9 .?A. 18. If?, m M T> If sees. 19. 2 3 88 sL. Exercises LXXIV. a. P. 3"- 1. 8-64. 2. 9-43. 3. 27-9. 4. 29-0. 5. 7-43. 6. 8-32. 7. 18-3. 8. 9-60. 9. 1-386. 10. 2-83. 11. 1-96. 12. 3-92. 13. 3-05. 14. 8-17. 15. 1-03. 16. 0-99. 17. 0-051. 18. 0-94. 19. 0-4986. 20. 3-006. Exercises LXXIV. b. P. 312. 1. (a) 31-1, (6) 20-2, (c) 78-8, (d) 104-6, (e) 158. 2. (a) 4-03, (b) 8-03, (c) 517, (a?) 203, (e) 40-4. 3. (a) 78-3, (6) 56-4, M "5-2, (d) 34-3. 4. 7-5 too small ( - ). 5. 7-5 too high ( + ). 6. o-oooi + . 7. 0-0000003 + . 8 - 4-04-. 9. 54 + . 10. 0-052-. 11. 43-- 12. 0-0368 + . 13. 0-126 + . 14, 0-63-. 15. 0-6 -. 16. 29-. 17. 100 -. 18. 0-02 -. Exercises LXXIV. c. P. 317. 1. (a) 4 i-9 , (6) 6, (c) 6-4, (d) 8-4. 2. 490. 3. (a) 87-93, (*) 144-6, (r) 291-6. 4. 97 miles nearly. 5. 49 yd. 6. 743 ft. nearly. 7. 38-4 ft. 8. 12 ft. 9. 188 yd. by 23-5 yd. 10. 7-3 in. 11. 2 ft. 4 in. 12. 5 ft. 11 in. 13. 35- 14- (a) 17 in., (b) 61 ft., (c) 26-6 ft., (d) 26-51 ft. 15. 12 ch. 21 Ik. 16. 9-80 cm. 17. 192 ft. v ANSWERS 637 Exercises LXXIV. e. P. 319- 1. (a) Final position 13-5 miles to E. and 14-1 to N., 19-5 miles between first and last position. 2. (a) 7-33 miles to E., 1-68 to N., distance 7-52 mi.; (b) 2-2 mi. W., 3-08 mi. N., distance 3-78 mi. ; (c) 11-6 mi. E., 4-3 mi. N., distance 12-37 mi. 3. (a) 0-17365, (b) 0-93969, (c) 0-50000, (d) 0-64279. Exercises LXXIV. f. P. 321. 1. 100-005. 2. 1200. 3. 300-01. 4. (a) 2^, (b) 2a, ( C ) 3 'd, (d) 5 th , (e) 3 rd , (/) 3 rd , (^) 3 rd - Exercises LXXIV. g. P. 322. 1. 8-12. 2. 25-01. 3. 40-5. 4. 49-22. 5. 32-25. 6. 55-3. 7. 15-36. ' 8. 10-5. 9. 44-36. 10. 42-65. Exercises LXXV. P. 324- 1. 7-69 tons. 2. 0-267 in. approx. 3. ^360. 8.r. 4^. 4. 3. 3*- 9^- 5 - 509-48 gr. 6. 253-2 min. 7. 175 Ib. 8. 10-625. 9. 1250 Ib. 10. 35 in. 11. 5447 Ib. 12. 31-62 ton. 13. 156-8 Ib. Exercises LXXVI. a. P. 329- 3. (a) J, (b) i (c) , (d) J, (*) ^V- 4. (a) 6 3 J, (*) 7iJ, W 76, (rf) 26j, (e) i8i, (/) 14. 5. (a) 6, (J) iii, (r) i7i, (af) 23!, (^) 30, (/) 37, (g) 44Y, (^) 53. Exercises LXXVI. b. P. 330- 1. (a) 1-05316; (b) 0-56004; (f) 0-33535; (<*) 0-45383 5 (e) 0-26624; (/) 0-59838; (,f) 0-21004; (A) 1-45220. 2. (a) 13 26' 23" ; (6) 47 W ; (f) S& 34' i" 5 (^) 178 42'. 3. (a) 34; W 0-504; (f) 78; (rf) 290-9; (e) 168; (/) 155. Exercises LXXVII. P. 331- 6. (a) 26, (6) 1 6, (^ 38, (aT) 20. 7. 1-7, 7, 6, 14, 62. 8. (J) 302 x io 6 , (c) 1 10 x io 6 . Exercises LXXVIII. b. P. 336. 2. 8^ cub. ft. 5. 132 sq. ft. 8. 2 in. 11. 9900 Ib. 3. ii ft. 6. 616 cub. ft. 9. -V- cub. ft. 12. 3 cm. 638 A MODERN ARITHMETIC 1. 8 sq. cm. 4. 7 ft. 7. 53900 c.c. 10. MP-^oz. Exercises LXXVIII. c. P. 337- 1. 754 c.c. approx. 2. 277 cub. ft. approx. 3. 152-4 x io cub. ft. approx. 4. 1706 cub. in. 5. i metre, 8-6 1 cm. 6. 0-04274... cub. ft. 7. i8381b. 8. 20-4 Ib. 9. 21-36 sq. ft. 1. (a) (i) 3583x10, (ii) 16-43 xio 4 ; (#) (i) ioi8xio 2 , (ii) 1264x10: (c) (i) 0-1257, (ii) 2-177; (d) (i) 2479x10, (ii) 1933. 11. o-O22cm. 12. 0-056 mm. 13. 1-025, 1-14 approx. 14. 0-25 cm. Exercises LXXIX. a. P. 338. 1. i, 8, 27, 64 c.c. 3. B 2, 2, 2, 8 ; C 3, 3, 3, 27 ; D 4, 4, 4, 64. 4. B 2 , C 3 3 , D 4 . 5. (a) 61076-64, (6) 7634580, (t) 954"3225. 6. 24ton6cwt. 7. ^(0-343). 8. I44O 3 . 9. 243 x-^. Exercises LXXIX. b. P. 340. 4. B 57 x io 6 , C 36 x io 6 , D 19 x io 6 , E 13 x io 6 . Exercises LXXX. a. P. 344. 1. 2ft. 2. 6-4 ft. 3. 2 days. 4. 9 days. 5. 42 cub. in. 6. 54 hrs. 7. 7200 8. 80%. 9. 20%. 10. i6oft. 11. 3-5- Ib. 12- 50%. 13. 4 years. 14- 5i%. 15. 30. 16. 30- 17. i6- 18. 22. Bs. 19. 42 | hr. 20. 1-28 hr. Exercises LXXX. b. P. 347- 1. (a] 745, (6) 27-1. 2. (), (), (/), (w) ; (), (/) ; (g) ; (0, (rf), (h), (i), (k). 5. 5-2 amp., 36 amp. hrs. 7. (a), (), (/) ; (3), (/), (c). 8- (-5)833, (')926, (flT)s6o, (e) 195, (/) 667. 9. 54 oz., ;f in. ANSWERS 639 Exercises LXXXI. P. 351- 1. 60. 2. 8-68 days. 3. 59. iSs. ?d. 4. 27 days more. 5. 3 hr. approx. 6. 3-49 hr. 7. 2-31 Cm. 8. 64. 9. loo. 10. 70 days in all. 11. 97-22 tons. 12. (a) 5 miles approx., (6) 24 ft. 13 - 93 A" da y s - 14 - () 2 ton s 19 cwt. I qr. 1 1 lb., (b) 40, (<r) ^40. 15. 7. 16. 62. 17. (a) 5-90 cm., (b) 0-0377 cm. 18. (a) 1-181 kilo, per sq. cm., (b) 12-35 c.c. 19 - 9^- 20. 21. 145. lid. 21. ^272.5^. 22. 201 -9 cub. ft. 23. (a) 3-2 ft. approx., (3) 752 lb. per sq. in. 24. 36 days. 25. 20 days. 26. 80. 27. (a) 155 ft. per sec., (b) 150 ft. 28. 19-91 ft. per sec. 29. 14-2 rain. 30. 71-7. 31- 573 kilo. 32. Countesses 1-04 times as thick as 'doubles.' S3. 262-5 days. 34. 265-68 sq. yd. 35. 1-29 mm. 37. 0-7556. 38. A 675, B 315, C 126. 39. A 163. &., B 28. 14*., C 39. 14*, D 38. 45. 40. 4. i2s. , 6. 41. igj., 24-r. 42. 60. 43. 97 sec. 44. 1.405x10* ft. 45. (a) 3-11 cm., (6) 301 grams. 46. (a) 2182 lb., (b) 1333^ lb. 47. 4 in. 48. 59-6 miles per hour. 49. (a) 9-8 sec., (b) 7-18 ft. Exercises LXXXII. P. 363. 1. rooo sq. Iks. 2. 1-9 sq. ch. 3. 5000 sq. Iks. 4. 4000 sq. Iks. 5. 7250 sq. Iks. 6. 2150 sq. Iks. 7. 8950 sq. Iks. 8. 5-35 sq. ch. Miscellaneous Problems LXXXIII. P. 365- 1. l6yT min. past 12. 2. (a) 27 j\ min. past 5, (b) i6 T 4 T min. past 3, (c) SfV min - to n. 3. (a) 43 IT mm - P ast 2, (<5) 5ii min. to 5, (c) l6i\ min. past 9. 4. i2f\sec. 5. 42/Vsec. 6. 8 | sec. 7. <a) IT, (6) T (0 THF. (^) 30 min. 8. 4^ min. 9. 70-18 sec. 10. 23-5 yd. 11. 42y rriin. past 6 ; 57- miles from A. 14. 334 days approx. 15. 15 cwt., 12-8 nearly. 16. II lb. of the cheap to each 8 lb. of the dear. 17. 26 : 19. 640 A MODERN ARITHMETIC 18. 38y\ min. past 10, 247^ miles, trains 35 miles apart, approach at 55 miles per hour. 19. 2 min., 880 yd. 20. 5! mile. 22. 15. 23. 3 T \ hr. 24. 32 min., 42 min. 25. 1800 yd., 9 ir| and gf m. per hr. 26. 2 miles, 24 min. after. 27. 60 days, 12 hr. 16 min. 28. 120 days. 29. i6yW days. 30. 5 days. 31. 8 days. 32. 5 days. 33. 30 boys. 34. 15^ past 24, I hr. 2^ min. 35. o.gs.S<t. 36. ^3.45. 37. 11:80. 38. 3 hr. 39. 5 miles. 40. B, 4! f ft. 41. 36 miles. 42. 3J gal. and 8 gal. 43. 8 : 9. Revision Exercises LXXXIV. P. 370. Miscellaneous. A 1. 324- 2. 30. gs. od. 3. 2 2 x3 2 x 13, 11x3x17, 7x13x17, 11x13x17; 7056. 4. 4. 5. 30-03, 4-4 kilo. 6. $. i>js. id. 7. 36-251, 1521. 8. lo^d. 9. 633 tons. B 1. 25. 2. 2-646. 3. ^126 4. 2 ft. 6. is. 8d. 7. 3 %. 8. 5662. 9. 6 cm. 4 mm. 10. 48-6261 sq. in., 313-7055 sq. cm., 6-451, true to units, units and 1st dec. figures respectively. 1. 2928. 2. 1255. 6*. 8rf. 3. 1. os. loaf. 4. 7%. 5. 45- 6- 6\/9~6, 4x^96. 7. 247-1. 8. 114. 9. (a) 24886, (6) 4-021. 10. 7749. D 1. 858357, remainder 48. 2. 20,^4.9^.2^. 3. (i) 2 T V; (ii) 7-038. 4. (i) 2 ton 17 cwt. 2 qr.; (ii) n miles 3 fur. 176 yd. 5. 218 fr. 51 cent. 6. 8fdl 7. 328. 145. grf. 8. 252-960. 9. 0. igs. od. approx. E 1. (a} 0-7275 ; (b) 26. igs. id. 2. (a) 295-502 ; (b) I3O97'53- 3 - 20 - 4 - 04. 5. 149, 1317160. 6. 7. ist cl. : 2nd cl. : : 9 : 10. 8. 6*. &/., 6*. io/., 6*. 5 ANSWERS 641 F 1. 4072. ;j. Sd. 2. 14400 3. i6J. 4. 160. 5. 951, 3487. 6. 2-2. 7. 3-280899. 8. T ^ T . 9. 3-4. 10. 75. 165-. 4</. nearly. G 1. 409! sq. ft, 46 sheets. 2. 5937^ cub. ft., 37109! gal. 3. 310284. 4. 8. 5. 1-55838. 6. 200%. 7. 1897, 1904. 8. 0. 5.$-. \d. 9. 6: i, 32. 10. 442-604, no decimal figure. H 1. 540, 3780. 2. |J. 3. 00000008562. 4. (i) 83-56; (ii) 14, 124. 5. 55-86591. 6. 135%. 7. 237-17. 8. 385. 9. 989 cub. in. 10. 2. gs. Qj. I 1. S.6s.SJ. 2. 3 and 1 1. 3. (i) -|g ; (ii) T WV 4. nooyd., loiooyd. 5. 1,1-0625,0-002,0. 6. 491 sq. m., sq. m. approx. 7. 42, 6, 25, 26. 8. 43911- I2j. 3</. 9. 358. 19*. 6dT. 10. i-32i5ac. J 1. ^547- i 7 s - 3d- % 3 2 7- 3 - 112 yd. 28 in. 4. i. 5. 32. 6. 0-0395, 0-006812. 7. 0-513, 8. 7373. 17.9. 7iflT. 9. 40^ bushels. 10. 224. 18.5-. 6d. 11. 47, 54. ,.,. . ,.29 o 19110 19110 19110 19110 19110 K L 5TJ7J- " " ""~" 3. 13 hrs. 32 min. 4. 27, 13. 5. i day 4 hr. 6. 1776 sq. yd., 1191 cub. yd. 7. 2f. 8. 0-028, 7-91 x io~ 3 . 9. 1 56. 10. 2940 miles. L 1. 9 cwt. i qr. 26 Ibs. 2. 200. 3. 3*. 4^. 4. 7630. 5. 292251960. 6. 6 mo. 7. 39-37 in. 8. _ I _ , 62-5 sq. in. 9. 529 x io 4 . 10. 52-9. M 1. 3-937- 2. 18. 17*. Sd. 3. 876576, 5218. 4. ioj hr. 5. 192 cub. yd., 208 sq. yd. ; w=g, = -,<=. 6. 420. 7. () Yes; () No; (0 No. 8. i. Ss. \\d., o. 2s. c&d. 9. 1658 mm., 3317 mm. N 1. 990 or 3366. 2. 6411,6835. 3. 603. 6*. 8</. 4. 12-8044. 5. (i) 58 sq. ft. 45 sq. in. ; (ii) 19 sq. ft. 99 sq. in. 6. 0-007782. 7. 44 ft. i in. 8. 20 ft. 9. 26-1%. 10. 9-1302. approx. J.M.A. 2S 642 A MODERN ARITHMETIC Exercises LXXXV. a. P. 389- 1. () 3 5 > (6) 4 6 , (0 io 11 , (</) 4 13 , W ioo 5 , (/) 25 63 , (^) < (0 -H*. (/) g)*^. 2. T 7 o- 3. T V 4. T V- 5. -f. 6. T V 7. io 13 io 3 ' Exercises LXXXV. b. P. 390 9. a*- 6 . 13. r 100 - 100 . Exercises LXXXV. c. P. 390. 1. 2-2. 2. ( rt ) IO- 3 , (3) IO 2 , (f) IO- 2 , (rf) ID" 4 . 3. (a) 5~ 4 , (^) 5 3 , (^) 5~ 3 . (e) $~\ (/) S' 6 - 4. (a) 2- 2 , (/;) 2- 3 , (c) 2~ 10 , (./) 2- 8 . 5. (a) 3-1, (3) 3 ~ 3 , (r) 3-6. 6. 2 and 3. 7. - I and - 2. 8. - 3 and - 4. 9. 2 and 3. 10. 5 and 6. 11. - 5 and - 6. 12. 5 and 6. 13. i and 2. 14. o and - I. 15. o and - I. 16. i and 2. 17. I and 2. 18. - 2 and - 3. 19. o and i. 20. - 2 and - 3. 21. I and 2. 22. - 5 and - 6. 23. - 6 and - 7. Exercises LXXXV. d. P. 391. 1. (a) 2, (b) -i, (c} -4, (d) 8, (*) 6, (/) -5, () io, (A) -2. 2. (a) i, (b) 3, (,) -i, (rf) 5, (,) -4, (/) -3, ( g ) 3. 3. (a) J, (b) 0-5, (0 -(0-5), (rf) -i. 4. (a) i, (J) -i, (c) -2, (d) 3, (') 4, (/) -5- 5. (a) -i, (3) 2, (r) 3, (d) -3, (') -2. 6. (a) 0-5, () 1-5, (c) -2, (rf) -2-5. 7. (a) 025, () 0-75, (^) 0-5, (d) -(0-5), (*) -1-25. 8. (a) -i, (b) -2, (c) 3, (rf) 2, (^) i, (/) 5, (#).-$. 9. (a)o-i = io~ 1 , (6) io 2 =ioo, (r) io 5 =iooooo, (d) 3 4 = 8i, (<f) 4-1 -5 = |- =0-1 25, (/) 2 10 =io2 4 , (^) io 3 = 0-001, (h} 5- 3 = y^- T = o-oo8, (i) x* = x*, ( 8= 4 , (^) 8=16^, (/) 16 = 8^. ANSWERS 643 Exercises LXXXV. e. P. 392. 1. (a) I and 2, (/;) 2 and 3, (c) o and I, (d) 4 and 5, (e) 5 and 6, (/) o and I, (g) i and 2. 2. (a) - 2 and - 3, (b) - 3 and - 2, (c) - 2 and - I, (d) - 5 and - 4. 3. (a) - I and o, (b) -2 and - I, (c) 3 and 4, (d) 2 and 3. 4. (a) 3 and 4, (b) o and - I. Exercises LXXXV. f. P. 394. 1. (c) 0-23 and 0-307, (d) 0-27 and 0-303, (<?) o and 0-5, (/) 0-466 and 0-533, (g) 0-43 and 0-478, (A) 0-47 and 0-53, (/) 0-71 and 0-855, (/) 0-846 and 0-92. 2. (a) 0-60206, (b) 0-69897, (<r) 0-77815, (a?) 0-90309. 5. 0-95424. Exercises LXXXVI. a. P. 399- 1. (a) 6-248 xio 5 gr., (6) 1-428 xio 5 gr., (<) 3-879 x io 5 gr., (d) 3-066 x 10 lb., (e) 0-4623 gr., (/) 337'9gr-, (g) i-56xio 4 lb., (A) i-533xio 10 lb. 2. (a) 1309, () 1093-6, (c) 725-9, (aT) 2703, (*) 733, (/) 2-319 xio" 5 , (g) 2-79 x io 4 , (A) 61-02. 3. (a) 43-5, (f) 4-58 5 (A) 9'95> (*) io-S- 4. (a) 35-88 c. ft, () 37-5 c. ft. 5. (a) 1269 gal., 0-056 in., (/;) 3807 gal., 0-168 in., (f) 5076 gal., 0-22 in. 6. (a) 0-007770, (b) 0-001551, (c) 0-001168, (d) 0-0005714, (e) 0-0004007, (/) 0-00003776. 7. 14924 lb. 8. (a) 0-0004525, 0-07539, (b) 0-094, 0-2953. (f) 0-088, 0-2764. 9. (a) 15-712, (b) 0-089607, (c-) 53-51. 10. (a) 1-2875x10, (0) 1-8345, (c) 0-38773, (</) 0-3650. Exercises LXXXVI. b. P. 403. 1. 96-43. 2. 0-00071 in. 3. 399 xio 2 . 4. (a) 170-2, () 3464, (c) 626 x io c. ft. 5. 7. (a) 3887, (b) 18-5. 8. (a) 32-9 ft., (b) 1759 lb. 9. 4-57 x io 2 . 10. (a) 1-96, () 12-3, (c) 49-2. 11. (a) 250, (b) 96-3 in. 12. (a) 3213 x io 2 , (b) 0-0047. 13. (a) 20, () 5, (<) 30. Exercises LXXXVII. a. P. 418. . 1. ^iooo(i-04) 3 . 2. ^5oo(i-02) 3 . 3. ^500(1-03)2. 4. i2oo(i-025) 4 . 5. ^8oo(i-03) 5 . 6. 7. ^9oo(i-o 4 ) 6 . 8. ^75o(i-o4) 6 (i-02). 9. 10. ^ 4 oo(i-04) 10 . 11. /40o(i-0325) 10 . 12. ^5oo(i-o5) 2 (i-o25). 13. ^750 (i-o6) 3 ( 1-03). 14. ^iooo(i-04) 4 (i-o2). 15. ^iooo(i-045) 4 (i-0225). 6 4 4 A MODERN ARITHMETIC Exercises LXXXVII. b. P. 419- 1. 5 00{(I.02) 6 -I}. 2. 500{(I-02) 7 -I}. 3. 4. iooo{(i-oo5) 12 -i}. 5. /2 6. iooo{(i-oi) 10 -i}. 7. i20o{(i-oo 5 p-i}. 8. Exercises LXXXVII. c. P. 420. 1. (<*) 153- 15*. <*/., (t>) 13. 13^., (') *9> & (00 38. 19-r. to the nearest shilling, (e) 72. s. (/) 126. 2J. o^, (^) 422. 18^. to the nearest shilling, (A) 2&3- 17-r- approx., (i) 11904 approx. 2. (a) 4-04, (A) 4-07, (c) 6- 168 approx., (d) 6-09. 3. Interest table : (a) (*) (f) (d) yrs. i p.c. a P.C. 2 p.C. *\ p.c. i I-OIOOOOO 1-0125000 O2OOOOO 1-0250000 2 I-O2OIOOO 1-0251562 O4O4OOO 1-0506250 3 1-0303010 1-0379707 0612080 1-0768906 4 I -0406040 1-0509443 0824322 1-1038129 5 1-0510105 1-0640811 1040808 1-1314082 6 1-0615202 1-0773821 1261624 1-1596934 7 1-0721354 1-0908494 1486949 1-1886857 8 1-0828567 1-1044850 1716788 1-2184058 9 1-0936853 1-1182911 I95II24 1-2488659 10 1-1046222 1-1322697 2190146 1-2800875 w (/) (g) yrs. 3 p.c. 4 P.C. 5 p.c. i 1-0300000 I -0400000 i -0500000 2 i -0609000 1-0816000 1-1025000 3 1-0927270 1-1248640 1-1576250 4 1-1255088 1-1698586 1-2155062 5 1-1592741 1-2166529 1-2762816 6 1-1940563 1-2653190 1-3400957 7 1-2298780 I-3I593I8 1-4071005 8 1-2667743 ' 1-3685691 1-4774555 9 1-3047775 1-4233119 1-5513283 10 1-3439108 1-4802444 1-6288947 ANSWERS 645 4. (a) 28. 17 j. orf., (6) 34. u. 6^., (t) 24. iar. 7^-, (^89. 9*. 8^., () 5 T 4- 14-r- approx., (/) 410. I2s. approx., (g) 34. 5^. gd. Exercises LXXXVII. d. P. 421. ! 5 yrs. 2. 3%. 3. 4%. 4. 1200. 5. 1219 approx. 6. 2j yrs. 7. (a) 1-005 %, (/;) 1-06 %, (r) 1-06 %. 8. 5-11, 5-12, 5-125. Exercises LXXXVII. e. P. 425. 1. (a) I, (b) uVs, (c) 8 or 2y|. 2. (a) 100, (b) 348. &. 4. (a) 853. or. saT., (/.) 2640. 15*. 6d. t (e) 507. igs. (d} 1507. 6j. 8^. 5. 1-966, 60 ft. 6. 70%. 7. 0-547 Ib. 8. 0-28 approx. 9- (<*) 3-0909, (6) 4-1836, ( f ) 5-3091. 11. 1154. los. 3d. 12. 1712. 8j. Oi/. 13. 7. 135-. zd. 14. 133. i8j. Sd. 15. Interest paid in each half-year instalment, ^"20, 18-174, J 6-3ii, 14-411, 12-472, 10-495, 8-478, 6-421, 4-323, 2-183. Principal paid 91-327, 93'i53> 95'Oi6, 96-917, 98-855, 100-832, 102-849, 104-906, 107-004, 109-144. Exercises LXXXVII. g. P. 430. 1. (c) 26 approx. 2. (*) 30 yrs., (/) 23, (g) 46 yrs. approx. 3. 172. 4. (e) 10. 55., (/) 21 yrs. nearly. 5. (a) 9, (b} 4-8 % approx. 6. (a) 1 8 yrs. approx., (b} 14 yrs. approx. 7. 8J yrs. approx. 8. 44^ yrs. approx. 9. (a) 6-4 yrs., (b) 17 yrs. approx. Exercises LXXXVIII. P. 434- 1. () 3-5, (*) 60, (<) 1710. 2. (a) 56, (0) 60, (c) 5280. 3. (a) 9, (b) 16, (0 336. 4. (a) 17-5, () 24, (f) 660. .5. (a) 6-5, () 24, (c) 924. 6. ( rt ) 5-5, (b) 12, (<) 126. 7- () 34-5, (^) 60, (c) 2730. 8. (a) 7, (b) 60, (r) 1080. 6 4 6 A MODERN ARITHMETIC 14. (a) 21751, (b) 2293, (c) 3588 xio 2 , (d) 6068-8, (e) 152850, (/) 791 A^. \U) 4 1 i/D 1 * V, ;.?) 3068 --Ml V X JO 2 , (A) 44212-8 (0 735368, * J~~^-9 \J 1 IJ'% (j) 3243 x 10% (/>) 1^64192, (/) 96510-7- 15. (a) I , :3 ' W 2 1:8, (f) 4 = 5, (rf) 9 = 7, (e) 16: 9, (/) * 2 : O/ 2 -* 2 ), 16. 1 , W J-7 t wjft ,W); 4_0_3_ 17. n-3 in. Exercises LXXXIX. P. 437 1. 2. 9J. 9^. 2. it I9-V. 9/. 3. ^o. 4*; lifl r 4. ^49- igjr. 4df. 5. ^72- O5". 4 6. 7- gs. 7^. 7. i. 6s. gd. 8. 13- 8s. i i\d. 9. 2. OS. irf. 10. 27. OJ. 2^/. 11. ji. us. 6 \d. 12. i- OS. ii F 13. 4- I3-T. 2^/. 14. 26. IOS. lid. 15. 2. 6s. n| a. 16. 4- 5-r. 4^</. 17. i. us. 4 18. 9- i& 19. 88'. 17*. Sd. 20. i. 3. r. offlil 21. Jan. 23, 1906 22. ^8 7 6 23. ^1250. 24. July i, 1905- 25. Sept . 10, 1905. 26. 3%. 27. 3%. 28. July i, 1905. 29. Jan. 30. 30. Jan . 8. 31. 2i%- Exercises XC. P. 440. 1. (a) I in., (b) ^15. 2. () 3-6 in., (b) 4-8 in. 3. (a) (i) 14-94, (ii) 1-25 ; (b) (i) 17-4, (ii) 2-25 ; (c) (i) 8-912, (ii) 1-25 (d) (i) 32-25, (ii) 16; (*) (i) 1 1 -4, (ii) 8-128; (/) (i) 12, (ii) 5 () (i) 60, (ii) II ; (h) (i) 15, (ii) 8; (0 (i) 20, (ii) 15 4. 188-5 sq. ft. 5. io sq. in (approx.). 6. 55-0 sq. in. 7. (a) 254-4 sq. in., (b) 235-6 sq. ft., (c) 175-9 sq. ft., (</) 189-4 sq. ft - (<?) 115-6 sq. ft. 8. 348-72 sq. in. 9. (2i-6ir) or 67-858 sq in. 10. 32 ITT. 11. 285 sq. in. (approx.). 12. () 757r or 235-62 sq. cm., (b) 104^ or 326-73 sq. cm., (c) i8gnr or 5940-7 sq. in., (d) 1771-- or 53-4 sq. in., (^) 26840^ or 843 x io 2 sq. in. Exercises XCI. a. P. 442. 1. 30-39 sq. in. 2. 475-3 sq. in. 3. 12-70 sq. cm. 4. 1932 x io 2 sq. cm. 5. 2865 sq. cm. 6. 1810 sq. in. 7. 3-801 sq. ft. 8. 7917 sq. cm. 9. 5542 sq. cm. 10. 42-7 cm. 11. 46-3 ft. 12. 5-1 in. 13. 10-3 cm. 14. 30-2 ft. 15. 20-1 cm. 16. 15-2 in. 17. 38-2 yd. I ANSWERS 647 Exercises XCI. b. P. 443- 1. 201 x io 6 sq. mis. 2. 44-178 sq. in. 3. 40-55 cm. 4. 50-265 sq. in., 0-160 Ib. 5. (a) 33-51 x io 6 , (6) 13-9 x io 6 , (c) 27-93 x io 6 , (aT) 50-8 x io 6 , (e) 39-4 x io 6 , (/) 64-88 x io 6 , (^) 5-586 x io 5 . 6. (a) 8-33 x io 6 , (b) 52-1 x io 6 , (<:) 80-2 x io 6 , (d) 30 x io 6 , (e) 100 x io 6 , (/) 2-75 x io 5 , (g) 1-27 x io 5 , (h) 14-46 x io 6 . 7. 407T or 125-7 sq. in. 8. 64^ or 201-06 sq. ft. 9. 160-758 sq. cm. 10. 163-363 sq. in. 11. 179-071 sq. in. 12. 1266-7 mm. 13 - 2264 Ib. Exercises XCII. P. 453- 1. (a) 44-12 cub. ft., (b) 136-19 cub. ft., (c) 9-2 in., (d) 7-5 cm. 2. 7-57 gal. 3. 4 gal. (approx.). 4. 422 gal. 6. 3 mm. nearly. 7. (a) surf 9161 sq. cm., vol. 825 x !O 2 c.c. ; (b) surf 16-62 sq. ft., vol. 6-37 cub. ft. ; (f) 128-7 sq. in., dia. 6-4 in. ; (d) surf 283-529 sq. in., dia. 9-5 in. ; (e) vol. 203-69 c.c., dia. 7-3 cm. 8. 14! cub. in. 9. 150-8 cub. in. 10. 53-4 cub. in. 11. 268 cub. in. 12. 396 cub. ft. 13. 180 grms. 14. 79-8 cub. in. 15. 9-45 Ib. 16. 4974 cub. ft. Exercises XCIII. P. 458. 1. (a) 1460 c.c., (b) 735c.c., (f) 3750 c.c. 2. 2600 cub. in. 3. 217 cub. yds. Revision Exercises XCIV. P. 459- Miscellaneous. A 1. 1325610. 2. 4-644 in. 3. 4. igs. oa. 4. ITT ac., 3465. 5. 5533. 6s. &/. 6. I T 4 T days. 7. 105. 8. i m. 4 dm. 9 cm. 8 mm. 9. 135 years nearly. 10. 0-26275 ac. B 1. 9rV 2 - ;5 2o8 3 approx. 3. 13-4. 4. 2 T. 18 cwt. 5. 47. 145-. zd. y 27s. iiJoT. 6. 495 cub. ft. 7. 204yd., A rain. 8. ^3-2^- 9- ^57, 6 in. roughly. 10. 2 yrs. 9 mo. approx. 648 A MODERN ARITHMETIC C 1. 23, 3-89 in. 2. Difference =0-0037 in. 3. 4. 3-14. 5. 11. 6. i. 6j. 8aT. 7. 866 approx. 8. 102-6 hrs. approx. 9. 29-85 ac. 10. 89304. D 1. 17810, 9-995, 0-40156625, 2120-2698. 2. 32%. 3. i. los. loiflT. 4. 120. 5. 2 yd. 6. 5. 7. 37. 4-r. orf. 8. 9. 1066. 135-. 4</. 10. 1-2 ac. approx. E 1. 3-141597, 1-00000127. 2. 2248. 6jr. 3<f. 3. 8 sees. 4- 3, 2. 85-. orf., o. 125. od. 5. 50 miles. 7. 53 approx. 8. 16 a shilling. 9. 130. \is.6d. 10. 0-247, 0-987, 2-221, 3-948, 6-169, 8-883, 12-091, 15-792. No. Yes. F 1. 7 cwt. 2 qr. 18 Ib. 2. 502386,505087,507788. 3. 1561-6. 4. 108. 5. 7TA/Z v ^ 6. $6168 x io 6 , $594 x io 7 , $4717 x io 6 , 62 x io 6 app. 7. ii mile. 8. 26-3 approx. 9. ij in. 10. 4 ac. 2 ro. 23 po. to nearest pole. Gr 2. 640 mm. 3. 161. <)s. 6d. 4. .453. IO.T. 3faT. 5. 4. i2s. g$d. 6. 27 miles. 7. 161-133 approx. 8 - i-59 % 9 - o. 6s. gd. 10. ^194. 45-. approx H 1. 3956. 2. 631. ns.'jd. 3. 21? %. 4. 1-53 m., 0-76 m. 5. |- liquid, -g 7 ^ water. 6. 0-0576723, 2-538 ft. approx. 7. 4^ %. 8. 24. 9. i : 0-993, : ' 7777- I. 36 ac. o ro. 23 po. approx. I ! lirf > 4-00001. 2. 101 -6 nearly. 3. 100%. 4. 192 in. 5. 119, 119 ft., 119 sq. ft. 6. 6 cub. ft. 192. cub. in. ; 4 yd. 7. 0-815. 8. 57. 65-. 4$a. 9. 10. ^280-07 J 1. (i) I -0004 nearly ; (ii) 0-7022 nearly. 2. 5-68 m., 1-602 sq. m. 3. 56. 4. 18-5625. 5. 24240 3rd, 9696 ist. 6. ^126. zs. od. 7. 715. 8. 3 o. 9. 3-495 xio 4 . 10. 4 ac. o ro. 25 sq. p. approx. K 1. 18524225. i cw. od. 2. 4%, 9. 3. (a) ii7^yd.; (6) 3! min. 4. 15 min. 16 sec. 5. 194-4 ton - 6. 14592 cub. in., 8064 cub. in., 6528 cub. in. 7. (a) 1503160; () 2000000. 8. i. 13*. 4^1 9. 146. 13-5-. ^d. 10. (a) 4 yr. 5^ mo. approx.; (b) 35^ approx. ANSWERS 649 L 1. 376, 379. 2. 20 casks, 21 mm. 3. 62-5. 4. 2-408, a\/~2. 5. 13. 6. 67-08 ft., 1 2- 1 ft. 7. i. igs. Sd. 8. 625. 9. 2:5. 10. 465 yd. and 846 yd. approx. M 1. I234f. 2. i min. 52^ sec. 3. 819. 6.r. id. 4. 25. 5. 2825761, 2560000. 6. 3550. 3124, 2982, amounts would be 3536, 3128, 3992. 7. 1000. 8. 20. 5-r. 4</., 2-09. 9. 18019, 25806, 9-055... ac. 10. 8, 40^ just outside C. N 1. I ro. 19 sq. po. 15^ sq. yd. 2. ii 2 x7 2 x I3 2 . 3. 54, 0-00096 gr. 4. 1210. 5. 8^ min. 6. 12 days. 7. 22hrs. 8. 27. 9. 11. os. Sd. 10. 1-16 ac. approx. Exercises XCV. P. 480. 1. 69 to nearest day. 2. 71 days to nearest day. 3. 83 days. 4. 22 ; Aug. 23. 5. (a) So, Sept. 20; (b) Aug. 15. 6. (a) 650, Nov. 25, 1906.; (b) Nov. 29, 1585 ; (c) 400, Dec. 3 ; (d) 1500, Mar. 7 in the following year. Exercises XCVI. P. 485- 1. (a) 7-98805. (b) 7-93787. 2. (a) 5-65518. (3) 9-44984. 3. (a) 3. 17*. io\d. (b) 4. 45-. iiar. 4. 3. 17*. 5^. 5. 935. 6. 132. 7. 0-941 in each case. 8. 0-678 approx. 9. 0-663 approx. 10. (a) o. gs. g\d. (b) 7s. n-^d. (c) 2. is. id. 11. (a) 0-90. (3) 0-98. (c) 0-917. (<a?) 0-900. (e) 0-874. (/) 0-900. ( t? -) 0-873. (A) 0-916. (*) 0-919. (/) 0-917. (k) 0-900. 12. 25.22. 13. (a) 1=6-305 roubles. (b) 1=9-762 yen. (c) 1 = 18-1595 kr. 14. (a) o-n fr. () 3927 fr. 15. 20-31 and 20-53. Exercises XCVII. P. 489. 1. (i) () 15,251- 17-s-. Z\d- (b} 15,233- i8j. 5K (0 I 5<349- 9 5 '. J f^- ' n 3 mo -5 or T 5, 2 34. 6j-. S^d. at present. (<^) X 5,337' 4 s - lid- m 3 mo ', or i5222. 4^. 4^. at present. 650 A MODERN ARITHMETIC (ii) () ;i73i- 4*. 6rf., (6) 1729- 3*- o^ (0 1742. $* (d) 1740. 175. 6^. (iii) (a) 2305. 125. irf., (6) 2302. 175. loaT., (r) 2320. 7s. id. (d) 2318. 25. l\d. 2. (a) 22,009. 4J- 2|dT. (0) 22,127. 3 J - IO K (r) 22,135. 75. 4/f. in 3 mo. (</) 22,168. os. 6d. in 3 mo. 3. 2i%. 4. 25-40. Exercises XCVIII. a. P. 494- 1- (a) i453. 25. 6aT., (*) 2124. I2J. 6d., (c) 1150. icw. orfl 2. (a) 2408. 15^. o</., (6) 4120. i or. <x?., (c) 4850. 3. (a) 440; (*) (i)6 9 o, (ii) 186.5^., (iii)20. Exercises XCVIII. b. P. 498. 1. 1131. 125. od. 2. 831. iu. 6d. 3. 1038. 155. od. 4. 22975. 5. 1084. 25. 6</. 6. 542. 185. 6^. 7. 37oi. 55. od. 8. 4440. 9. 459. 75. 6rf. 1- ^9837. 105. od. 11. 1254. 85. od.' 12. 5812. 105. od. 13. (a) 3-046, () 3-503^ (f) 3-429, (^) 3-483, M 3-333, (/) 3-670, (^r) 3-431, W 3-315, (0 3-352, (/) 4-124, (/&) 3-9025, (/) 3-465, (;//) 3-449, (n) 3-297, (0) 3-390. 14. (a) 45. 85. o</. gain, (<$) 109. 135. od. loss, (<r)i23. iO5.o^.loss, (</) 126. 175. od. gain, (*) 4. 25. od. loss, (/) 37. 25. od. loss, (") ^ Io - IOS - d- S a ' n , (^) 29. 155. od. loss, (z) 2. 25. od. loss, (/) 424. 185. od. loss. 15. 16. (a) 5 , (*) 5 (r) 3 , (rf) 2 T , (*) S (/) 6, (-) 2. 17. (a) 4, 5^, (6) 3. 4J. 6^., 2^, 8, (r) 6. i6j. 4^., 5^, 7^, (d) 3, 6, (e) 3> 5^. 7, (/) 2f , 4, 6. 18. 300000. 19. 22000, 10%. 20. 1650. 21. 1487. los.od. 22. 510. 23. 81141 1, 34485. 24- 357. 2J. i a/., 1642. 17^. 2^., 1785. 45. 3</. 25. (a) 326925, (/;) o. i&r. ioa. 26. (a) 1139500, (<J) 1-449^. 27. 5%. 28. 3 ii57approx. 29. 8433331. 30. 1197023. 31. 11538. 32. 9727. 33. 753951. 34. 5-62. ANSWERS 651 Exercises XCIX. P. 510. 1. (a) 5 hr. 25 min. (b) 4 hr. 30 min. (c) 5 hr. 13 min. 2. (a) 5 hr. 37 min. (t>) 5 hr. 53 min. (c) 5 hr. 38 min. 3. (a) 4 hr. 33 min. (b) 250 and 255 min. (c) 4 hr. 15 min. 4 hr. 1 8 min. 4. (a) 4 hr. 26 min. (b) 3 hr. 52 min., 3 hr. 58 min., 4 hr. 10 min., 4 hr. 30 min. (c) 4 hr. 22 min. 6. Mode 4 hr. 15 min. 4 hr. 30 min.; 5 hr. 45 min. 6 hr.; 4 hr. 4 hr. 15 min.; 3 hr. 50 min. 4 hr. 5 min. 7. (a) 23-67. (b) 23-65. (c) 23-655. (a) s. d. s. d. 8. Arith. Av. i . 19 . 5-4 2 . 8 . IO| 3-4 Median i . 19 . 8 ii 2 . 8 . 10 ii 3 . 4 . 8 10 Mode i . 19 . 03 2.8.4 s. d. s. d. 3.4. 33 -4-5 3.4. 63 . 4 3.5-6-3.5. Arith. Av. Median Mode * d. 4. ii . 3 4. ii . i 4. ii . o<| to I 4. ii . 6j s. d. 19 . n 18. ii 19. 5 2 . C to 3. c 9. (a) 10. (a) (c) (e) (g) (0 11. (a) (c) (i) 6 st. ij Ib. and 6 st. 2 Ib. (i) 45-63. (ii) (i) 23-65. (ii) (i) 42-83- (ii) (i) 29-4. (ii) (i) 7-41. (i>) (i) 100109. (i) I30-I39-9. (i) 90-94-9. (i) 95-4- (i) 80. 46-647-1. 22-8. 42-942-2. 29-6-29-8. 7-2- (ii) 105-7. (") "8-2. (ii) 97-7. (ii) 95-4- (ii) 90. (<) 6 st. \\ Ib. (ii) 36-636-7. (ii) 29-5. (ii) 1 1-8 12-2. (") 3-2. (ii) 25-426. (b} (i) 120129-9. (ii) 129-4. (d) (i) 95-99-9- () 99-i- (/) (i) 96. (ii) 92. (h) (i) 120. (ii) 120. (b) 6 st. to 6J st. (*) (i) 3578. (d) (i) 28-72. (/) (i) 12-01. (A) (i) 3-18. (/) (i) 26-42. 652 A MODERN ARITHMETIC 10 / \ 13 11 _5 _3 ? ,,171 .38 -779 ,,33 12. (a) 4^0, 4iT, 3s> 3*. 2 $> 3Too> 3T5> 2 iroo> 24^. (/;) 3-64 true to 2nd dec. place. (r) 3 3-5. (d) 33. 13. (i) (a) 93^ 92j, 9i2 9 o, 92^i, 95f > (*) 94-2. (0 92-593- ( d \ 93- (ii) (a) 1884-1888, 1,847,240; 1889-1893, 9,015,740; 1894-1898, 4,371,420; 1899-1903, 8,353,120; (b) 6,141,900 to nearest 100 ; (c) mode in groups of 1,000,000 between 2,000,000 and 3,000,000; (rf) 5,780,600. (a) S9 f 5, S9 JJ, 59i 6o/<j, 6iJJ, 6i|J, 6of J, (b) 53-9281. (c} 60-4. (d) 60 2". (e) 59 15. Value from arithmetic mean 30-12 30-24. 16. (a) 22 23. (b) 25-6 25-8. (c) 26-0. 17. 10, 0-633 cm. 18. (a) 32-0432-026. (b) (37oo2)xio- 8 (c) 492-57490-23. 19. (a) 49*. nd., 88-1 %. (b) 49*. lid., 61-6%. (c) 2^s. 3^,48-0%. Exercises C. P. 524. 1. (a] 24. (b) 120. (<r) 720. 2. (a) 24. (<$) 210. (c) 120. 3 - H950- 4. (a) /e- W SF- (f) A- 5. 1365. 6. 12150. 7. Exercises CI. P. 528. ! () 7- 13*- 2aT. (*) 7. 5J. iorf. 2 - () 7- 19^- 3- (*) ^7- I", "fl?. 3. (a) 9. 8*. la?. (6) 8. 17.9. irfl 4. (a) 5. 8j. 7^. (*) 5. 2s. gd. 5. (a) 14. i7j. irf. (^) 13. 9,. 6. i. 3*. 8rf. 7. i. 9*. 8</. 8. 2. 5*. fd. 9. 2. IQJ. 2d. 10. 3. 9^. 3^. 11- i. 5^. 6^1 12. 3. i8j. 5^. 13. (a) 18-93. (*) 18-31- (0 I7-7I- (^) 17-12. () 20-68. (/) 20-01. ( ff ) 19.38. (y^) 18-66. (a) 631. IQJ. yd. (b} 1066. 3*. gd. (c) 1293. i8j. 4^. -r- () 95^- oj. 6flT. ANSWERS 653 Revision Exercises Oil. P. 529. Miscellaneous. A 1. 16836, 2 3 x 3 2 x 5 x 1 1 x 37 = 146520. 2. (i) 0-253037 ; (ii) o. 14*. i id. 3: 39-32. 4. 47. 2s. iod. approx. 5. 33 min. past I. 6. 10 hr. 48 min. 7. 16-7. 8. 0-9. 9. 1552. IQJ. <*/., 26347. i or. od. 10. 2-89 miles from C. B 1. 2 3 x3 3 x7 3 xii 3 . 2. 453-1. 3. 12 miles 936 yd. 4. 2i2j ft. 5. 241. 5*.. 6d. y 308. 6s. od., 290. 8j. 6d. approx. 6. 17. i or. oaT. 7. 1-02. 8. 5. 2*. 7%d. 9. 115. 95-. gar. nearly. 10. a= -41-6, = 11-24. 1. 496. 2. 112 sq. yd., i. i6s. od. 3. 3-3227. ^- 35 O ' *. 1 1 336 to nearest sov. 6. 4331. 7. 17. 8. ?r(36+ 14-^13). 9. i. IOT. 8|flT. per oz., 8. 10. 44, 44. D 1. 19- 2. 417000. 3. 62 Ib. 6-8 oz. 4. 17. IDS. od,, 70, 157. iof. od. 5. 3. 3j. 8of. 7. 5-56 sq. mile. 8. 15. 9. 14400, 83 . 10. 9. E 1. (i) S6fS 5 (ii) 924. 2. 2 x 13x41, 3x 5x ii 2 , 2 3 x3X79, 2 3 x 3 x 5 x ii 2 x 79 x 13x41. 3. 20526 cub. ft. 171 cub. in. 4. 600 to good, less 13. Js. 6d. bank charges. 5. 77. i is. od. 6. 2-09787 kilo. 7. 2 hr. 47 min. to nearest min. 8. 4\/3~ or 6-928 in. 9. ( 100 + ), 35. 4250. 10. 2-646 in. F 1. 0-508857. 3. 703, 1234455346. 4. 81097 sq. mile. 5. 48-64. 6. No, I.T. 3jflT. 7. 5-7596 cub. ft., 323-95 Ib. 8. 1912. 95. 9irf. 9. 23- 14-f- orfl 10. (i) 8 h. 9^ m. a.m. ; (2) 3 hr. 46 m. ; (3) Dec. 21. G 1. (i) 891-4; (ii) No. 2. C 8531. 3*. orf. ; D 5687. Ss. 8d. 3. 9370-0739, 570-73 sq. ft. 5. 2049. 3^- T d - 6. 22-5 yr. approx. 7. 7. 8. 10. 14^. 6d. 9. 37 ft. 6 in., 30 ft., 12 ft. 654 A MODERN ARITHMETIC HI. 21-13859 Ib. 2. 2-34186. 3. (a) 173-155; (*) 18-891. 4. 621. los. 5. (a) 48-8 in. ; (b) 20-95 in. 6. 8-31. 7. 2306. 5.1-. od. 9. 2-49 cm. approx. I 1. 23. 2. 0-318. 3. 43, 45. 4. 10 increase. 5. 21-8%. 6. 14. 15*. 4fcflT. 7. 381 J cub. in., 8. 1:17.568. 9. 88Sff, 44 f f, 5 55FT- 10. 2 ac. 3 ro. 21 po. approx. J 2. if^. 3. V> alPoV- 4 - 3*- 5 - 428-456 kilo. 6. i333. 6j. 8aT., 3Te %. 7. 122-5 cub - in -> 2 4"5 cub. in. 8. 487. idr. &jT. 9. 38 class A, 62 class B approx. 10. 31 ac. o ro. 33 sq. po. K 1. 2-78,2.718282. 2. 105 xio 6 . 3. i j. 3</., MIX io 6 approx. 4. 4-06294. 5. (i) 0-04; (ii) 0-27. 6. 13-1 in. 7. 2604.3^.4^. 8 - 61:35. 9 - I 0'5- 10 - 60,92-5,123. L 2. (a) 44; (6) 30.55-. 6-6d ; (c) 82-6234; (d) 1-5241. 3. 714 cub. in. 4. 50. 5. 0-84. 6. 4%. 7- 49 % gold, 51 % silver. 8. 0-75 ft. 9. Diminished, 30 : 29. I M 1. 0-27959 approx. 2. 43-2. 3. 8. i6s. gd. 4. 12. 5. 10,49-8; 36x1 miles per hour. 6. Tax, 12. i6s. od.; income, 243. 45-. od. 7. 7-6 gal. 8. 84 years. 9. 440 approx. 10. I : i nearly. N 1. 410 ft. 2. 10182 Ib. 3. 305267^- 4. 461 ft. 5. 23 % nearly. 6. 1 1 miles per hour. 7- 23-37, 25-87, 25-7, 28-54, 33-34, 26-98. 8. 3. i3j. gd. 9. 3750. 10. (i) 4-58, 0-22 ; (ii) 3-55. 1. o. u. Sd. 2. 0-900. 3. I02f|-. 4. 195, 3480 5. 9231-6 cub. ft. 6. Second expression. 7. 103 cm., 53 cm. approx 8. 26-5 per thousand. 9. 642 approx. 10. Loses 1.05-. lod. onioo P 1. 92 Ib. 2. 84. 3. |f, 4201819314. 4. 42 Ib. 5. ^f approx., 16-68. 6. ?^ 4SOOO> 7 - 33T . 27, 22! days respectively. 8. 3000. 9. T 7 o- mile, y mile. 10. 0-87 ft., 0-78 in. ANSWERS 655 Typical Examination Papers. P. 555. PART I. Cambridge Local (Preliminary}. I. 1. 1176009, One million one hundred and seventy-six thousand and nine. 2. 14407. Ss. 10^. 3. 62. I2s. Q\d. 4. jC'j. gs. ^d. 5. 77. 6. (i) J-, (ii) T V 7. 7-014. 8. o. 17.?. lod. 9. 16366-. 30 c. 10. 12. y. od. 11. ,54. 12. Preliminary. II. 1. 494078. 2. 15983- u. W. 3. 87. &. iijar. 4. 336. 5. (i) |, (ii) |. 6. 2-12. 7. 1265. 8. 2 fr. 80 c. 9. 23. Ss. gd. 10. o. u. 7. t 18-4%. 11. 2^ miles. Preliminary. III. 1. 120005. 2 - ;7999- 19-r- 8aT. 3. ^365. I2J. 6^. 4. 321. 5. 435. is. vt. 6. (i) 6, (ii) TV : ?. 75800, 0-00758. 8. 6. isj. a/. 9. 2588 sq. yds., 23292 sq. ft. 10. 8 kilom. 47 m. 11. 10. Junior. A. 1. 19. 2. 441000 sec. 3. 1,167960. 4. 11. iu. qd. 5. 73- or. loiflT. 6. 326 fr. 4 c. 7. 3. i;j. 7^. B. 1. 34-56. 2. 14. 3. A 200, B 328. 4. 192. 5. 2600 dollars. 6. 92. 656 A MODERN ARITHMETIC Oxford Local (Preliminary}. /. 1. 103. los. n\d. 2. 4752. 3. 71. 4. 0-4448. 5. 0-84375. 6. T f o- 7- 47- 7-r. 10* 8. 16. 9. 2\. 1. 2257- ifa. 8; 4. i. 14.5-. 4* 7. 14. i6s. 9fd Preliminary . II. Id. 2. 38. 5. 0-3015. : s. ^1000. 3. 6f|, 39?. 6. 0-0375. 9. 146. 6s. of* Prelim inary. HI. 1. ,15. 75-. i^oT. 2. 2, 7, 13, 14, 26, 91, 182. 4. -aV 5. 0-080325. 6. J. 7. 14. 8. 17. 9J. 6ar. 9. 3^. Preliminary. Higher Arithmetic. 1. 0-0187. 2. Costal. 12 j. 6</. 3. 3^ years. 4. 36 marks 1 1 pfennigs. 5. Each son^iooo, each daughter 600, each nephew ,200. 6. Increase of ^35. 7.^11.5^.0^. Junior. 1. if. 2. T V_ 3. 96. 14*. 4-125* 4. -Vy~ = 2-236 1 1 . . . , \/5 = 2-236068. 5. 44 francs 8 centimes. 6. 13 men. 7. ^"163. is. 6d. 8. 0. 185-. 5* 9. 400. 10. 5% at 132. College of Preceptors, Third Class. I. 1. (i) 34138054, (iii) seventeen thousand four hundred and eighteen. 2. 142 tons 6 cwt. o qr. 24 Ib. 3. 6 ac. 3 ro. 37 sq. po. 14-75 sq. yds. 4- 7475- IOT. o* 5. 29 T 1 2 9 <j, irV 6. I. 7 - T(T' Tff> 2 T> 1 2^0' 8. 0-2I402, 1200. 9. 4. gs. u|* 10. ^156. 15^. 4^* 11. 9216. Professional Preliminary. 1. 384- 2. i. i 9J . 6* 4. 5. 0-861406. 6. 0-243 nearly. 7. 17 fr. 72 c. 8. 134. 2-r. 8* 9. 46cm. 10. 5911 yds. 11. jn. 13^-. 4*, 16. 13^-. 4* 12. 322. ANSWERS 657 Second Class or Junior. I, 1< ;i953- 2J- & 2. 49. 3. 9, 57 4. 21 min. 34 sec. 5. (i) 696^! , (ii) i\\. 6. io-ioiioi. 7. 42-28, 23810-03325. 8. ^1776. 5*. oaf. 9. 3 . ioj. od. 10. 0-0000035,0-00025. 11. Sunday, 8a.m., 540 knots. 'Ihird Class. II. ! 30. 2. 8 hrs. 3. 4. I 3 j. 6* 4. 419.^.21* 5. H- 6. jfo. ? "S1F TT77T) ;- I-?- 6^. 8. 0-00400437, 0-3016. 9. 2000. 10. 0. 13-5-, 4^. ' 11. 1815 tons. Second Class or Junior. II. * 28 73- 2. 192. i2s. 6d. t ^195. i 3 j. u|^., 2952. 3. 1848 ; i, 24, 28, 42, 44. 4. 2 . 5. 0-144,0-017,0-4585. 6. 2 days 5 hr. o min. 49 sec. 7. ji. 8j. 4^. 8. ^41. u. 9^. 9 - (i) TerW> (ii) 0-002086. 10. 27yJ, ^o. gj. ioaH 11. 0-075 cm. ///. 1. 129413 + 59. 2. ;i2. i8j. 7J<* 3. I2tons2cwt. 4. ^62. or. 8^. 5. |. 6. 7 ||. 7. 2T- 8. 0-06045, 1374. 9- 44- 1- A 5625, ^1875, B ^3750. ^1250, C 1875, ^"625, D ^3750, 1250. 11. 1296. Pupil Teacher Candidates' 1 Examination (Junior Grade}. l - 4T4 % 2. 43, 0-526 in. 3. jo. ior. zoj^ 4. ^2. 9J. 7aT. 5. 0. 2f. 44^. Pupil Teacher Candidates' Examination (Senior Grade). 7. 9 tons 13 cwt. 2 qr. 8. ||. 9. 48. 10. 162 sq. ft. 11. ;o. i2s. 6d. 12. jg. 13^-. od. Teachers' Certificate Preliminary Examination, Part I. 1. 3 years 38 days. 2. 96. 3. 7-35. 4. 1. &. 2far., 1. &. 3* 5. 2. &r. 9i/. 6. 252. 7. 1566. 8. 110885897421. !0. 99i6. I 3 j-. 4^7. 12. o. IQS. gd. 13. 3 % per annum. 14. 98 Ib. 7 oz.; 113. 15. (a) 10000, (6) ^2395, (c) 3, (^) ^4. 6f. ftd, J.M.A 2T 658 A MODERN ARITHMETIC Typical Examination Papers. P. 571. PART II. Civil Service. I. 1. 864. 2. 3. 3. 60-617, 0-127, 0-945, 64: i, i : 7-4 4. 4-3, 212, 28 Ib. 5. \od. 6. 1:4-7; 12-7 cm.; 93 approx. Civil Service (Higher Arithmetic"). 1. ^"134. 5s. od. 2. 1 170 approx. 3.2^. 4. 42 % approx. 5. 2. i$s. od., 3. 3*. od., 2. i7s. od., 9-5 %, 19 % 52 Ib. 7. 85 gal. approx. 8. 15^ min. from the start, 17872 ft. Certificate Examination, 1. i, 3, 4. 4. 70-4 ft. per sec., 88 ft. per sec., 352 ft., 440 ft. 5. 19-635 sq. cm. 6-438 sq. dm. 6. (a) 125, (6) T V> fc) 2999080. 8. 166-5 miles, 10 min. past 1 1, to nearest minute. College of Preceptors. \st Class (Mensuration). 2. ii ac. i ro. 3. ^"166.6^.0^. 4. 9-5 chains. 5 - 7- 15*. %\d. 6. 182 yd. 7. 1130. 8 - ^"25. 13.?. 4d. 9. 864 + 268^2. 10. 7-24 ft. Ib. approx. ' 11. o 84 84 o 35i 305 205 71 o 129 ; 0-472095 ac. o Cambridge Local. Junior (Mensuration). 1. ,27. ioj. Qd. approx. 2. n in. approx. 5. 18-1 sq. ch. London Chamber of Commerce. 1. 2-68 cm. 2. 266f%. 3. 175 Ib. 4. (a) ;35-3i875> (6) o. 17-r. 5 Jar., (;) ^48103. QJ. 4 flT. 5. 502. IOJ. o</. 6. 50-175, 61-3, 0-05, 0-6. 7. 3 tons 6 cwt. 108 Ib. 8. 258-7 approx. 9. ^298. 105. 6d. approx. Oxford Local (Senior). 1. 2, 3, 7, 11, 66, 77, 154, 231. 2. 0-21805. 3. 225 ft. 4. 307 blocks, 48828. 5. ^547.17^.2^. 6. 4.48p.m. 7. 14%. 8. 2 litres, 12 litres. 9. 5:13. Cambridge Local (Senior). 1- ;33- 3-r- <* 2. 47. 3. f. 4. 1213 ft.; 0-925. 5. 141 fr. 12 centimes. 6. 122. TS. $d. 7. ^148.7^.3^. 8. 24%. 9. ^42. 10. Time 5.13 jf p.m.; Distance from A 2 if miles, INDEX PART I (The numbers refa to pages} Addition, 4 ; by steps, 8 ; of long columns, 10 ; compound, 56 ; of length, 79 > of metric length and decimals, 91 ; of fractions, 166. Approximations, 236. Approximate or contracted multipli- cation, 259. Area, 116; British measures of, 117; metric, 118; measurement of irregular, 126. Averages, arithmetical, 40. Capacity, measurement of, 136 ; j liquid measures, 138 ; dry mea- sures, 138 ; problems on, 139. Compound addition, 56 ; subtrac- tion, 58 ; multiplication, 61 ; division, 65. Contracted division, 268 ; multipli- cation, 259. Cubed numbers, 26. Decimal coinage, 130. Decimalization of money, 239 ; possible errors, 243. Decimals, introduction of, 88 ; addition and subtraction, 91 ; multiplication and division by a single figure, 95 ; division of, general case, 107 ; division of, to a given number of decimal places, no ; multiplication of, 98. Decimals as vulgar fractions, 169 ; recurring, 171. Division, 29; by factors, 31 ; general method, 34 ; Italian, 35 ; true to nearest unit, 36 ; compound, 65 ; contracted, 268 ; of decimals, by a single figure, 95 ; of decimals, general case, 107 ; to a certain number of decimal places, ill ; lengths, 79 ; metric lengths, 95 ; of vulgar fractions, 172. Equivalent decimals to a vulgar fraction, 169 ; tractions, 162 ; graphic illustrations, 165. Errors in data, 254 ; in decimalizing money, 243. Factors, 37. Fractions, 157 ; graphic illustrations of, 159; improper, and mixed numbers, 160 ; addition and sub- traction of, 1 66 ; equivalent, 162 ; equivalent, graphic illustrations of, 165 ; expressed as decimals, 170 ; multiplication and division, 172 ; one quantity as a fraction of another, 161 ; square root of, 278. Greatest common measure, 182 ; graphic illustration, 186 ; general method, 188. Interest, 278 ; simple, 283. Least common multiple, 191; general method, 192. 66o A MODERN ARITHMETIC Length, addition, subtraction, multi- plication and division, 79 ; British measures, 74; (contimied}, 113; metric, 75 ; reduction, 75 ; the measurement of, 74. Metric system, of length, 75.; reduc- tion, 85 ; addition, 91 ; multipli- cation by a single figure, 95. Money, British systems, 53 ; deci- malization of, 239 ; foreign, deci- mal coinage, 130. Multiplication, 18 ; by a single digit, 21 ; factors, 22 ; general case, 22 ; compound, 61 ; contracted, 259 ; of decimals, general case, 107 ; fractions, 172; lengths, 79; metric, 95. Notation and numeration, I. Practice, 244. Percentages, 257. Proportion, 197. Recurring decimals, 171. Roots, square, 273. Scale drawing, 220. Signs, use of, 97. Simple interest, 283. Squared and cubed numbers, 26. Square root, 273 ; general method, 274 ; of vulgar fractions, 278. Subtraction, 12 ; compound, 58 ; of fractions, 1 66. Time, measurement of, 147. Volume, 132; tables of, 134. Weight, measurement of, 141 ; apothecaries', 142 ; avoirdupois, 142 ; metric, 142 ; troy, 142. INDEX 661 PART II ( The numbers refer to pages} Angles, measurement of, 327. Approximations, in division, 309 ; in multiplication, 307 ; in square root, 314, 320. Areas of circles, 331 ; conical sur- faces, 436 ; rectilinear figures, 358; triangles, 432 ; spherical surfaces, 441. Averages, 505. Banker's discount and bills of ex- change, 436. Bills of exchange and Banker's dis- count, 436. Chain, use of the, 303. Circles, areas of, 331. Combinations and permutations, 521. Compound interest, 414 ; compound interest law (geometrical pro- gression), 421 ; tables, 573. Cones, areas of, 438 ; volumes of, 447. Cylinder, 334. Days, table O M 572. Discount (Banker's), 436. Division, approximations in, 309 ; by logarithms, 398. Equation of payments, 478. Evaluation of formulae by loga- rithms, 402. Evolution by logarithms, 399. Examination papers (typical), 555. Exchange, bills of, 436 ; foreign, 487. Expectation of life, 524. Foreign bills of exchange, 487. Geometrical progression, 421. Graphic method of finding square roots, 318. Interest, compound, 414; tables, 573. Inverse proportion, 342. Involution, by logarithms, 399. Length, measurement of, 297. Logarithms, introductory, 387 ; determination of logs. etc. , 393 ; division, 398 ; evaluation of for- mulae, 402 ; evolution, 399 ; involution, 398 ; multiplication, 398 ; use of logarithms, 396. Method of mean squares, 510. Money exchange. 482. Multiplication, approximations in, 307 j by logarithms, 398. Payments, equation of, 478. Permutations and combinations, 521. Prism, 334. Probabilities, 521. Proportion, 323 ; inverse, 342 ; examination for, 347. Pyramid, volume of, 447. 662 A MODERN ARITHMETIC Reciprocals, table of, 569. Rectilinear figures, areas of, 358. Revision exercises, 370, 459, 529. Screw gauge, 297. Similar figures, 338. Slide rule, 407. Sphere, area of surface of, 441 ; volume of, 450. Spherometer, 297. Square root, approximations in, 314, 320 ; graphic method of finding, 318 ; table of, 569. Statistics, 505. Stocks, 491. Tables, compound interest logarithms, 570 ; reciprocals 569 ; days, 572 ; square roots, 569- Triangle, area of, 432. Vernier, 299. Volumes of, cones and pyramids, 447 ; frusta, 448 ; graphic calcula- tion, 454 ; spheres, 450. , ds, its. GLASGOW: PRINTED AT THE UNIVERSITY PRESS BY ROBERT MACI.EHOSE AND co. LTD. A MODERN ARITHMETIC BY H. SYDNEY JONES, MA. Part /., separately, 2s. 6d. SOME PRESS OPINIONS. School World. " It is stated in the preface to this book that in it an attempt has been made to provide a course of work in which the essential parts of arithmetic are dealt with in the spirit of recent recommendations. The attempt seems to us to be very successful. The expository parts of the book are clear and reasonable, while the exercises are exceedingly varied, free from unnecessary complications, and yet of sufficient difficulty to call forth the best efforts of the average pupil." Schoolmaster. " Teachers on the look-out for a suitable book for their elder scholars will do well to examine this. It contains all the necessary rules from the earliest work to square root and simple interest, together with graphic and practical exercises. The tests for revision are exceedingly well selected." Teacher's Aid " The attention of teachers is strongly drawn to a book so replete with intellectual methods and with sound common sense." Teacher. "A really excellent volume. . . . We know no better book of its type." Educational News. " This is one of the most meritorious publications in this department for some time. It aims at (and successfully) producing a clear conception of units, accuracy, quickness, and reasoning power. . . . For teachers the graphic parts of the work are of the greatest value, and the book being most moderate in price, deserves soon to be universally used." LONDON : MACMILLAN AND CO., LTD. A MODERN ARITHMETIC BY H. SYDNEY JONES, M.A. Part //., separately, 2s. 6d. SOME PRESS OPINIONS. Schoolmaster. "With the cunning hand of the master, Mr. Jones applies the finishing touches. The ordinary text-book teaches arithmetic ; this shows how practical use can be made of it. From cover to cover the book is thoroughly good, and will delight many a one who has a fair knowledge of the higher rules but does not see how to apply it" Educational News. "With the utmost confidence we recommend this Arithmetic to all teachers of mathematics. They will find in it much that is new, and at the same time necessary for present-day requirements." School World. " Part II. fully maintains the high standard reached in Part I. The treatment is singularly fresh and instructive; it is thoroughly practical, and at the same time intelligent, providing an excellent discipline while avoid- ing the dangers of too great abstraction. . . . We very cordially recommend it to the consideration of teachers. " Guardian." There is evidence of careful thought and experienced discretion in the arrangement and choice of material." School Guardian. "The volume will be found to meet not only the require- ments of the more conservative examining bodies, but also those of the Army, Civil Service, and other examiners keenly alive to the change which is taking place in mathematical teaching." LONDON: MACMILLAN AND CO., LTD. YB l7234 >/ / ^ Y ^hsu *{ , P^ ^^ OA THE UNIVERSITY OF CALIFORNIA LIBRARY f